history of prime numbers assignment

history of prime numbers

With the Roman conquest of the Greeks, much of the written Greek knowledge was translated to Latin, or at least preserved. As the Greeks taught the Romans what they knew, they preserved Greek mathematical knowledge but made no further progress in the study of pure mathematics, such as prime numbers.

The Arab mathematicians of the Middle Ages studied the work of ancient Greek mathematicians but with the added advantage of a numerical system more amenable to computational work. Thabit ibn Qurra , for example, proved the relationship between consecutive prime Thabit numbers and amicable pairs .

By this time, the most common definition of prime number was “a number that is divisible by 1 and itself.” 1 fits this definition, but some mathematicians were troubled by the ways in which 1 is different from the other prime numbers. Euler, for example, was motivated by the fact that σ ⁢ ( p ) = p + 1 for prime p but not for p = 1 to regard 1 as not a prime number. Others were no more bothered by 1’s peculiarities than those of the number 2, such as Moritz Stern , who did not consider 3 a Stern prime because he considered 1 prime.

In the 20th century, computers gradually became important in calculating data for theorists to ponder; from the 13th Mersenne prime on all the largest primes since the middle of the century have been found with the help of computers. The invention of public key cryptography in the late 1970s has precipitated the need for larger prime numbers and motivated many advances in integer factorization algorithms. From the 1990s onwards, distributed computing projects like the Great Internet Mersenne Prime Search and Seventeen or Bust have discovered some of the largest known prime numbers.

• 1 Wells, D. The Penguin Dictionary of Curious and Interesting Numbers , London: Penguin Group. (1986): 20
• 2 Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics , Penguin Group. (2004)

.1 External links

PrimeFan, http://geocities.com/primefan/Prime1ProCon.html Arguments for and against the primality of 1

 MacTutor

Prime numbers.

• The Twin Primes Conjecture that there are infinitely many pairs of primes only 2 apart.
• Goldbach's Conjecture ( made in a letter by C Goldbach to Euler in 1742) that every even integer greater than 2 can be written as the sum of two primes.
• Are there infinitely many primes of the form n 2 + 1 n^{2} + 1 n 2 + 1 ? ( Dirichlet proved that every arithmetic progression : { a + b n ∣ n ∈ N } \{a + bn | n \in \mathbb{N}\} { a + b n ∣ n ∈ N } with a , b a, b a , b coprime contains infinitely many primes. )
• Is there always a prime between n 2 n^{2} n 2 and ( n + 1 ) 2 (n + 1)^{2} ( n + 1 ) 2 ? ( The fact that there is always a prime between n n n and 2 n 2n 2 n was called Bertrand 's conjecture and was proved by Chebyshev . )
• Are there infinitely many prime Fermat numbers? Indeed, are there any prime Fermat numbers after the fourth one?
• Is there an arithmetic progression of consecutive primes for any given ( finite ) length? e.g. 251 , 257 , 263 , 269 has length 4 . The largest example known has length 10 .
• Are there infinitely many sets of 3 consecutive primes in arithmetic progression. ( True if we omit the word consecutive. )
• n 2 − n + 41 n^{2} - n + 41 n 2 − n + 4 1 is prime for 0 ≤ n ≤ 40 0 ≤ n ≤ 40 0 ≤ n ≤ 4 0 . Are there infinitely many primes of this form? The same question applies to n 2 − 79 n + 1601 n^{2} - 79 n + 1601 n 2 − 7 9 n + 1 6 0 1 which is prime for 0 ≤ n ≤ 79 0 ≤ n ≤ 79 0 ≤ n ≤ 7 9 .
• Are there infinitely many primes of the form n # + 1 n\# + 1 n # + 1 ? ( where n # n\# n # is the product of all primes ≤ n n n . )
• Are there infinitely many primes of the form n # − 1 n\# - 1 n # − 1 ?
• Are there infinitely many primes of the form n ! + 1 n! + 1 n ! + 1 ?
• Are there infinitely many primes of the form n ! − 1 n! - 1 n ! − 1 ?
• If p p p is a prime, is 2 p − 1 2^{p} - 1 2 p − 1 always square free? i.e. not divisible by the square of a prime.
• Does the Fibonacci sequence contain an infinite number of primes?

References ( show )

• B C Berndt, Ramanujan and the theory of prime numbers, Number theory Madras 1987 ( Berlin, 1989) , 122 - 139 .
• V N Chubarikov, Problems in prime number theory that are related to classical theorems of P L Chebyshev, Moscow Univ. Math. Bull. 46 (5) (1991) , 15 - 19 .
• H Cohen, Les nombres premiers, La recherche 26 (278) (1995 . ) , 760 - 765 .
• L E Dickson, History of the Theory of Numbers (3 volumes ) ( New York, 1919 - 23 , reprinted 1966) .
• U Dudley, Formulas for primes, Math. Mag. 56 (1) (1983) , 17 - 22 .
• U Dudley, History of a formula for primes, Amer. Math. Monthly 76 (1969) , 23 - 28 .
• J Echeverria, Observations, problems and conjectures in number theory-the history of the prime number theorem, in The space of mathematics ( Berlin, 1992) , 230 - 252 .
• L J Goldstein, A history of the prime number theorem, Amer. Math. Monthly 80 (1973) , 599 - 615 .
• A Granville, Harald Cramér and the distribution of prime numbers, Harald Cramér Symposium, Scand. Actuar. J. (1) (1995) , 12 - 28 .
• S Das Gupta, The story of prime number, Ganita Bharati 16 (1 - 4) (1994) , 37 - 52 .
• F Ischebeck, Primzahlfragen und ihre Geschichte, Math. Semesterber. 40 (2) (1993) , 121 - 132 .
• F Manna, The Pentathlos of ancient science, Eratosthenes, first and only one of the 'primes' ( Italian ) , Atti Accad. Pontaniana ( N.S. ) 35 (1986) , 37 - 44 .
• L E Mauistrov, Prime values of the polynomial x 2 +x+ 41 ( Russian ) , Istor.-Mat. Issled. 27 (1983) , 63 - 67 .
• O Ore, Number Theory and Its History (1948 , reprinted 1988) .
• J Pintz, On Legendre's prime number formula, Amer. Math. Monthly 87 (9) (1980) , 733 - 735 .
• P Ribenboim, The little book of big primes ( New York, 1991) .
• P Ribenboim, The book of prime number records ( New York-Berlin, 1989) .
• W Schwarz, Some remarks on the history of the prime number theorem from 1896 to 1960 , in Development of mathematics 1900 - 1950 ( Basel, 1994) , 565 - 616 .
• R de La Taille, Nombres premiers : 2000 ans de recherche, Science et vie 838 (1987) , 16 - 20 , 146 .
• H S Uhler, A brief history of the investigations on Mersenne numbers and the latest immense primes, Scripta Math. 18 (1952) , 122 - 131 .
• A Weil, Number Theory: An Approach Through History from Hammurapi to Legendre (1984) .

Additional Resources ( show )

Other pages about Prime numbers:

• A History Topic on Perfect numbers
• Legendre's estimate for π ( n )
• Gauss's estimate for π ( n )
• Legendre and Gauss estimates compared

Other websites about Prime numbers:

• Great InterNet Mersenne Prime Search
• University of Tennessee USA
• Bertelsen's number and mistakes in calculating π ( n )

Funding for this project generously provided by Overdeck Family Foundation

History of Mathematics

A prime number is a positive integer having exactly one positive divisor other than 1. Prime numbers are therefore the building blocks of counting numbers and were studied as early as 250 BCE. In modern times, the study of the distribution of prime numbers has led to some of the most beautiful and famous results in mathematics, large primes play an important role in encryption and the search for ever-bigger primes continues to challenge the limits of advanced and distributed computing.

around 200 BCE

The Sieve of Eratosthenes

Finding the primes by crossing out everything that isn’t

Cataldi's Divisor Table

Oldest divisor table, giving divisors for numbers up to 750

Guldin's Factor Table

One of the oldest factor tables, giving factorizations up to 10,000

Van Schooten's Prime Table

Prime table up to 10,000

SWAC Computes New Mersenne Primes

Five new Mersenne primes

Turing's Zeta Function Machine

Riemann zeta cogs

Why prime numbers still fascinate mathematicians, 2,300 years later

Associate Professor of Mathematics, University of California, Santa Cruz

Disclosure statement

Martin Weissman receives funding from the Simons Foundation for collaboration in mathematics.

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On March 20, American-Canadian mathematician Robert Langlands received the Abel Prize , celebrating lifetime achievement in mathematics. Langlands’ research demonstrated how concepts from geometry, algebra and analysis could be brought together by a common link to prime numbers.

When the King of Norway presents the award to Langlands in May, he will honor the latest in a 2,300-year effort to understand prime numbers, arguably the biggest and oldest data set in mathematics.

As a mathematician devoted to this “Langlands program,” I’m fascinated by the history of prime numbers and how recent advances tease out their secrets. Why they have captivated mathematicians for millennia?

How to find primes

To study primes, mathematicians strain whole numbers through one virtual mesh after another until only primes remain. This sieving process produced tables of millions of primes in the 1800s. It allows today’s computers to find billions of primes in less than a second . But the core idea of the sieve has not changed in over 2,000 years.

“A prime number is that which is measured by the unit alone,” mathematician Euclid wrote in 300 B.C. This means that prime numbers can’t be evenly divided by any smaller number except 1. By convention, mathematicians don’t count 1 itself as a prime number.

Euclid proved the infinitude of primes – they go on forever – but history suggests it was Eratosthenes who gave us the sieve to quickly list the primes.

Here’s the idea of the sieve. First, filter out multiples of 2, then 3, then 5, then 7 – the first four primes. If you do this with all numbers from 2 to 100, only prime numbers will remain.

With eight filtering steps, one can isolate the primes up to 400. With 168 filtering steps, one can isolate the primes up to 1 million. That’s the power of the sieve of Eratosthenes.

Tables and tables

An early figure in tabulating primes is John Pell, an English mathematician who dedicated himself to creating tables of useful numbers. He was motivated to solve ancient arithmetic problems of Diophantos, but also by a personal quest to organize mathematical truths. Thanks to his efforts, the primes up to 100,000 were widely circulated by the early 1700s. By 1800, independent projects had tabulated the primes up to 1 million.

To automate the tedious sieving steps, a German mathematician named Carl Friedrich Hindenburg used adjustable sliders to stamp out multiples across a whole page of a table at once. Another low-tech but effective approach used stencils to locate the multiples. By the mid-1800s, mathematician Jakob Kulik had embarked on an ambitious project to find all the primes up to 100 million.

This “big data” of the 1800s might have only served as reference table, if Carl Friedrich Gauss hadn’t decided to analyze the primes for their own sake. Armed with a list of primes up to 3 million, Gauss began counting them, one “ chiliad ,” or group of 1000 units, at a time. He counted the primes up to 1,000, then the primes between 1,000 and 2,000, then between 2,000 and 3,000 and so on.

Gauss discovered that, as he counted higher, the primes gradually become less frequent according to an “inverse-log” law. Gauss’s law doesn’t show exactly how many primes there are, but it gives a pretty good estimate. For example, his law predicts 72 primes between 1,000,000 and 1,001,000. The correct count is 75 primes , about a 4 percent error.

A century after Gauss’ first explorations, his law was proved in the “prime number theorem.” The percent error approaches zero at bigger and bigger ranges of primes. The Riemann hypothesis, a million-dollar prize problem today, also describes how accurate Gauss’ estimate really is.

The prime number theorem and Riemann hypothesis get the attention and the money, but both followed up on earlier, less glamorous data analysis.

Modern prime mysteries

Today, our data sets come from computer programs rather than hand-cut stencils, but mathematicians are still finding new patterns in primes.

Except for 2 and 5, all prime numbers end in the digit 1, 3, 7 or 9. In the 1800s, it was proven that these possible last digits are equally frequent. In other words, if you look at the primes up to a million, about 25 percent end in 1, 25 percent end in 3, 25 percent end in 7, and 25 percent end in 9.

A few years ago, Stanford number theorists Robert Lemke Oliver and Kannan Soundararajan were caught off guard by quirks in the final digits of primes. An experiment looked at the last digit of a prime, as well as the last digit of the very next prime. For example, the next prime after 23 is 29: One sees a 3 and then a 9 in their last digits. Does one see 3 then 9 more often than 3 then 7, among the last digits of primes?

Number theorists expected some variation, but what they found far exceeded expectations. Primes are separated by different gaps; for example, 23 is six numbers away from 29. But 3-then-9 primes like 23 and 29 are far more common than 7-then-3 primes, even though both come from a gap of six.

Mathematicians soon found a plausible explanation . But, when it comes to the study of successive primes, mathematicians are (mostly) limited to data analysis and persuasion. Proofs – mathematicians’ gold standard for explaining why things are true – seem decades away.

• Mathematics
• History of science
• Prime numbers
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2 Prime numbers: The DNA of mathematics

• Published: January 2020
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The Pythagoreans developed many of the ideas related to numbers that have become so familiar to us, including even and odd numbers, square numbers, triangular numbers, and so on. They also discovered that some integers cannot be decomposed into factors. These are called prime numbers and they constitute the building blocks of all the other integers, called composite. This chapter deals with the prime numbers in a general non-technical way, since much of the writing about them is quite specialized. The prime numbers have remarkable properties, many of which are still resistant to being proved. Prime numbers matter deeply to mathematics, not to mention to the progress of human knowledge generally. Pythagoras believed that prime numbers were part of a secret code which, if deciphered, would allow us to unlock the mysteries of the cosmos itself.

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• Mastering Prime Numbers: A Student's Handbook for Math Excellence

Prime Numbers, Prime Help: A Student's Handbook to Math Assignment Excellence

Mathematics, often perceived as a formidable subject, possesses a unique ability to challenge students across diverse educational levels. Amidst the vast array of mathematical topics encountered by students, prime numbers emerge as particularly intriguing and fundamental entities. Serving as the focal point of this handbook, the aim is to provide a guiding light for students navigating the intricate realm of prime numbers, imparting not only valuable insights but also equipping them with essential problem-solving strategies. The overarching goal is to foster a comprehensive understanding that goes beyond mere memorization, thereby laying a solid foundation for achieving excellence in math assignments. As students delve into the depths of prime numbers, the handbook becomes a compass, steering them through historical perspectives, unveiling the properties of primes, and introducing powerful identification techniques like the Sieve of Eratosthenes and modern primality testing algorithms. The journey extends to the application of prime factorization in real-world scenarios and explores the enigmatic world of number theory with a focus on Goldbach's Conjecture, twin primes, and prime gaps. Mathematical proof techniques, such as proof by contradiction and mathematical induction, are demystified, offering students valuable tools to fortify their logical reasoning skills. If you need help with your math assignment , this handbook serves as a comprehensive resource to enhance your understanding and problem-solving abilities in prime numbers and related mathematical concepts.

Beyond the theoretical aspects, the handbook bridges the gap between prime numbers and practical applications, delving into the pivotal role they play in cryptography, exemplified by the RSA algorithm, and their broader implications for data security. Venturing into the natural world, the guide uncovers the connections between prime numbers and the Fibonacci sequence, unraveling the patterns within primes that manifest in the structures of nature. Culminating in strategies for excelling in prime number assignments, the handbook provides not only problem-solving techniques but also a curated selection of resources and tools to empower students on their mathematical journey. In essence, this handbook encapsulates the essence of prime numbers, transforming them from abstract concepts into tangible tools for academic success. As students navigate the complexities of prime numbers armed with knowledge, insights, and problem-solving skills, they are poised not just to meet the challenges of their assignments but to develop a profound appreciation for the beauty and significance of mathematics in their academic journey.

1: Unveiling the Mysteries of Prime Numbers

Unveiling the Mysteries of Prime Numbers takes us on a captivating journey into the heart of one of mathematics' most fascinating subjects. In this chapter, we delve into the fundamental definition and properties of prime numbers, laying the groundwork for a comprehensive understanding. Prime numbers, those indivisible integers greater than one, have intrigued mathematicians for centuries, and this section aims to demystify their enigmatic nature. Exploring the historical perspective, we traverse the annals of mathematical discovery, uncovering the profound contributions of luminaries like Euclid, Eratosthenes, and Euler. Their collective efforts shaped the understanding of primes, illustrating their crucial role in the development of number theory. Moving beyond history, this chapter introduces the Sieve of Eratosthenes, a powerful tool for identifying prime numbers efficiently. Students are guided through the step-by-step process of sieving out primes, gaining practical insights into this ancient yet timeless algorithm. The chapter also touches upon modern primality testing methods, shedding light on probabilistic and deterministic approaches. By the end, students emerge equipped with a foundational understanding of prime numbers, their historical significance, and practical tools to discern these elusive digits, setting the stage for a journey deeper into the world of mathematical excellence.

1. Definition and Properties

Prime numbers, the building blocks of mathematics, are integers greater than 1 with no divisors other than 1 and themselves. Their properties, such as infinite existence and unique factorization, set them apart in the realm of number theory. Understanding their significance forms the basis for exploring their applications and unraveling the mysteries they hold in various mathematical domains.

2. Historical Perspective

Embarking on the journey of prime numbers requires a glimpse into history, where ancient mathematicians like Euclid and Eratosthenes laid the foundation for understanding these elusive digits. Their contributions, along with subsequent luminaries like Euler, mark the evolution of prime number theory. As we unravel historical perspectives, we gain insights into the fascination and challenges that have shaped our comprehension of prime numbers throughout the ages.

2: Identifying Prime Numbers

Identifying prime numbers is a crucial skill in mathematics, and it forms the bedrock for various mathematical concepts and applications. One of the fundamental tools for this task is the Sieve of Eratosthenes, a historical algorithm that efficiently sifts through numbers to reveal the primes. It operates on the principle of elimination, systematically marking multiples of each prime until only the primes remain. This method not only showcases the simplicity and effectiveness of ancient mathematical techniques but also provides students with a hands-on approach to understanding the essence of prime numbers. In the realm of modern mathematics, primality testing algorithms take center stage, allowing for the swift identification of prime numbers through probabilistic and deterministic methods. These algorithms play a pivotal role in cryptography, where the security of systems relies on the properties of prime numbers. By comprehending these identification techniques, students not only grasp the theoretical aspects of prime numbers but also acquire practical skills applicable to real-world problem-solving. The chapter on identifying prime numbers aims to empower students with the tools necessary to confidently recognize and work with these essential mathematical elements, laying a solid foundation for more advanced mathematical pursuits and contributing to a broader appreciation for the intrinsic beauty of prime numbers in the mathematical landscape.

1. Sieve of Eratosthenes

Equipping yourself with a powerful tool for prime number identification is essential in the journey to mathematical excellence. The Sieve of Eratosthenes, a centuries-old algorithm, offers an efficient method for sieving out prime numbers. This ancient technique, attributed to the Greek mathematician Eratosthenes, provides students with a practical understanding of how to systematically identify and generate prime numbers, laying a crucial foundation for further exploration into the world of number theory and mathematical problem-solving.

2. Primality Testing

In the modern landscape of number theory, primality testing algorithms play a pivotal role in determining whether a given number is prime or composite. This chapter delves into the intricacies of these algorithms, distinguishing between probabilistic and deterministic tests. Students will gain insights into the practical applications of these methods in various real-world scenarios, showcasing the relevance and importance of primality testing in cryptography, computer science, and beyond. Understanding these testing techniques empowers students to confidently navigate the complexities of prime numbers and apply their knowledge to solve mathematical problems with precision and accuracy.

3: Prime Factorization

Prime factorization, a fundamental concept in number theory, is the process of expressing a given integer as the product of its prime factors. This chapter delves into the elegant simplicity of the Fundamental Theorem of Arithmetic, which states that every positive integer greater than 1 can be uniquely represented as a product of prime numbers. Understanding this theorem is akin to unlocking the mathematical DNA of numbers. By exploring the intricacies of prime factorization, students not only gain a deeper insight into the structure of numbers but also develop essential problem-solving skills. The applications of prime factorization extend beyond the realm of theoretical mathematics; they are pervasive in various fields. From cryptography, where it forms the backbone of secure encryption algorithms, to practical everyday scenarios like simplifying fractions and finding common denominators, prime factorization plays a crucial role. This chapter not only guides students through the step-by-step process of prime factorization but also illuminates the real-world implications of this mathematical concept. As students grasp the power of prime factorization, they not only enhance their mathematical proficiency but also gain a tool that transcends the boundaries of the classroom, proving to be a valuable asset in diverse mathematical and practical situations.

1. Fundamental Theorem of Arithmetic

Dive into the elegance of prime factorization through the Fundamental Theorem of Arithmetic. This theorem asserts that every positive integer can be uniquely represented as a product of prime numbers. Serving as the cornerstone of number theory, it unveils the inherent order within the realm of integers. By breaking down numbers into their prime constituents, mathematicians gain a powerful tool for understanding and manipulating numerical relationships. The theorem's simplicity belies its profound implications, shaping various mathematical principles and laying the groundwork for more advanced concepts in the intricate landscape of prime numbers.

2. Applications of Prime Factorization

Explore the practical applications of prime factorization, transcending the abstract realm of mathematics into the tangible world. This fundamental concept, derived from the unique representation of numbers as products of primes, finds utility in diverse fields. From cryptography, where it forms the basis of secure encryption algorithms, to simplifying complex fractions in everyday arithmetic, prime factorization proves itself as a versatile and indispensable tool. This chapter unveils the real-world relevance of prime factorization, showcasing its omnipresence and effectiveness in solving problems beyond the confines of mathematical abstraction.

4: Prime Numbers in Number Theory

In the realm of number theory, the exploration of prime numbers opens a gateway to profound mathematical inquiries. Chapter 4 delves into the intriguing landscape of prime numbers within the context of number theory, unraveling mysteries and presenting challenges that have captivated mathematicians throughout history. Goldbach's Conjecture takes center stage, proposing that every even integer greater than 2 can be expressed as the sum of two prime numbers. This longstanding unsolved problem not only showcases the intricacies of prime numbers but also exemplifies the tenacity of mathematical curiosity. Twin primes, pairs of primes with a difference of 2, contribute to the rich tapestry of number theory, inspiring ongoing research into their distribution and patterns. The exploration of prime gaps, the spaces between consecutive primes, unveils both the regularities and irregularities in the prime number sequence, posing questions that propel mathematicians into deeper investigations. As students traverse this chapter, they gain not only a conceptual understanding of these mathematical phenomena but also an appreciation for the intellectual challenges embedded in the study of prime numbers within the broader scope of number theory. The intricate relationships and unsolved conjectures within this realm beckon students to embrace the beauty of mathematical exploration, fostering a mindset that seeks both answers and further questions in the captivating landscape of prime numbers and number theory.

1. Goldbach's Conjecture

Delve into the fascinating realm of number theory with Goldbach's Conjecture, a hypothesis proposing that every even integer greater than 2 can be expressed as the sum of two prime numbers. This centuries-old puzzle, named after the German mathematician Christian Goldbach, has intrigued minds with its simplicity and elusiveness. Despite its apparent simplicity, proving this conjecture for all even numbers remains an open problem, contributing to the allure and ongoing research in the field of number theory.

2. Twin Primes and Prime Gaps

Embark on a journey exploring the captivating world of twin primes and prime gaps, essential aspects of prime number distribution. Twin primes, pairs of primes with a difference of 2, and the study of prime gaps, the spaces between consecutive primes, unveil both patterns and mysteries in the realm of number theory. Investigate the intriguing question of whether there are infinitely many twin primes, a query that continues to drive mathematical inquiry. The study of twin primes and prime gaps provides a window into the distribution of prime numbers, revealing both order and complexity in the mathematical landscape.

5: Mathematical Proof Techniques

In the vast landscape of mathematical exploration, Chapter 5: "Mathematical Proof Techniques" serves as a crucial juncture for students aiming to grasp the intricacies of prime numbers. At the forefront of this chapter is the art of proof, a fundamental skill in the mathematician's toolbox. The section opens with a deep dive into the proof by contradiction technique, an elegant method that plays a pivotal role in establishing the primality of numbers. By guiding students through the logical steps involved in assuming the negation of a statement and subsequently deriving a contradiction, this section equips learners with a powerful approach to tackling complex mathematical problems. Additionally, the chapter unfolds the significance of mathematical induction, a technique that enables the systematic verification of statements for an infinite set of numbers. As students navigate through illustrative examples and practical applications, they develop a nuanced understanding of how these proof techniques form the backbone of mathematical reasoning. Moreover, the mastery of these proof techniques not only enhances one's ability to validate conjectures and theorems related to prime numbers but also instills a broader skill set applicable across various mathematical domains. Ultimately, Chapter 5 acts as a gateway for students, propelling them towards a deeper appreciation of the logical foundations underpinning prime number theory and fostering the analytical mindset essential for success in mathematical pursuits.

1. Proof by Contradiction

Master the art of mathematical proof with the proof by contradiction technique. This powerful method involves assuming the opposite of what you want to prove, leading to a logical contradiction. Applied extensively in number theory, particularly in establishing the primality of numbers, proof by contradiction enables students to systematically demonstrate the validity of mathematical statements. By embracing this technique, students enhance their analytical skills and gain confidence in constructing rigorous proofs, a valuable asset in tackling complex problems and advancing their understanding of mathematical concepts.

2. Mathematical Induction

Explore the formidable realm of mathematical induction, a technique widely used in proving statements about prime numbers and beyond. This method involves establishing a base case and demonstrating that if a statement holds for any given case, it also holds for the next case. Students will grasp the elegance of this technique as it provides a systematic and logical approach to solving problems with recurring patterns. Mathematical induction is a versatile tool, equipping students to tackle complex mathematical challenges and fostering a deeper appreciation for the interconnectedness of mathematical concepts in the realm of prime numbers and beyond.

6: Prime Numbers and Cryptography

In Section 6, we delve into the enthralling intersection of prime numbers and cryptography, unraveling the intricate role these elusive digits play in securing modern communication systems. At the heart of this exploration lies the RSA algorithm, a cornerstone in the world of data security. As we embark on this cryptographic journey, we unravel the layers of the RSA algorithm, dissecting how prime numbers act as the linchpin for its effectiveness. The RSA algorithm leverages the difficulty of factoring large composite numbers into their prime components, exploiting the mathematical complexity inherent in this process to encrypt and decrypt sensitive information. By comprehending the mathematical intricacies of prime numbers, students gain insights into the robustness of cryptographic systems and the formidable challenges posed to potential attackers. Beyond the algorithm itself, Section 6 also illuminates the broader applications of prime numbers in data security. From facilitating secure key exchange to underpinning digital signatures, prime numbers emerge as the unsung heroes safeguarding the confidentiality and integrity of information in the digital realm. This section not only equips students with a practical understanding of cryptography but also instills a sense of appreciation for the pivotal role prime numbers play in the evolving landscape of data protection. As the digital age continues to advance, the significance of prime numbers in cryptography remains paramount, making this exploration an essential cornerstone for any student aiming to excel in the intricate world of mathematical applications in data security.

1. RSA Algorithm

Step into the realm of cryptography and discover the pivotal role prime numbers play in securing modern communication systems. Explore the RSA algorithm, a widely used encryption technique, and understand how prime numbers are at the core of its security. From secure key exchange to digital signatures, grasp the significance of prime numbers in ensuring the confidentiality and integrity of sensitive information.

2. Applications in Data Security

Uncover the broader applications of prime numbers in data security, from secure key exchange to digital signatures. Grasp the significance of prime numbers in ensuring the confidentiality and integrity of sensitive information. As the foundation of cryptographic protocols, prime numbers contribute to secure communication, protecting data from unauthorized access and manipulation. Their use in algorithms guarantees the robustness of encryption methods, making them indispensable in contemporary data security practices.

7: Prime Numbers in Nature

The chapter on "Prime Numbers in Nature" unravels the fascinating interplay between prime numbers and the intricate patterns found in the natural world. One prominent connection lies within the Fibonacci sequence, where the occurrence of prime numbers unveils a mathematical harmony evident in various aspects of nature. As students delve into this section, they will explore the mesmerizing relationships between prime numbers and the Fibonacci sequence, witnessing their manifestation in the structures of plants, animals, and celestial bodies. Beyond the Fibonacci sequence, the chapter sheds light on the intriguing patterns within prime numbers, such as the Ulam spiral and the distribution of primes along number lines. These patterns, seemingly abstract at first glance, reveal themselves as integral components of the natural order. The chapter emphasizes the beauty of mathematical patterns in the world around us, encouraging students to appreciate the elegance of prime numbers as they appear in the formations of leaves, petals, and even the galaxies. By understanding the inherent connection between prime numbers and nature, students not only gain a deeper insight into mathematical concepts but also develop a heightened awareness of the mathematical principles that govern the universe. This exploration of primes in nature serves as a bridge between theoretical mathematics and the tangible wonders of the world, illustrating the pervasive influence of prime numbers in shaping the very fabric

1. The Fibonacci Sequence

Explore the fascinating connection between prime numbers and the Fibonacci sequence, a naturally occurring phenomenon in various aspects of nature. The mathematical beauty behind these relationships manifests in the spirals of shells, the arrangement of leaves, and the growth patterns of plants, showcasing the intricate link between the Fibonacci sequence and primes. This chapter unravels the mysteries of nature's numerical language, providing insights into the underlying mathematical principles governing the world around us.

2. Patterns in Primes

Delve into the intriguing world of patterns within prime numbers, where mathematical regularities emerge in unexpected ways. From the mesmerizing Ulam spiral to the distribution of primes along number lines, these patterns offer a glimpse into the profound beauty inherent in the sequence of prime numbers. This chapter explores the exploration of mathematical structures, revealing the interconnectedness of primes and unveiling the intricate patterns that continue to captivate mathematicians. The study of patterns in primes not only deepens our understanding of number theory but also underscores the elegance and complexity embedded in the fabric of mathematics.

8: Strategies for Excelling in Prime Number Assignments

Strategies for Excelling in Prime Number Assignments," students will find a treasure trove of practical insights and techniques tailored to enhance their proficiency in dealing with prime number-related tasks. This segment aims to empower learners with problem-solving strategies that go beyond mere memorization, fostering a deeper understanding of the underlying concepts. By emphasizing systematic approaches, students will learn to break down complex problems into manageable components, identify recurring patterns, and employ appropriate mathematical tools with precision. The section further provides guidance on effective time management, enabling students to tackle assignments with efficiency and confidence. As students delve into the intricacies of prime numbers, they'll discover the significance of employing proof techniques like contradiction and induction, fostering critical thinking skills essential for mastering mathematical concepts. Additionally, the section offers a curated selection of resources and tools, ranging from online platforms to interactive simulations and textbooks, providing students with valuable assets to augment their learning experience. Armed with these strategies, students will not only navigate prime number assignments more effectively but also develop a holistic approach to problem-solving that extends beyond the realm of mathematics, laying the foundation for academic excellence and a lifelong appreciation of the intricate beauty inherent in the world of prime numbers.

1. Problem-Solving Techniques

Mastering prime numbers demands effective problem-solving techniques. A systematic approach is essential, identifying patterns and applying appropriate mathematical tools. This chapter provides students with the skills needed to tackle complex assignments. From leveraging mathematical induction to employing proof by contradiction, these techniques empower students to unravel the intricacies of prime numbers with confidence, fostering a deep understanding of mathematical concepts.

2. Resources and Tools

Excel in prime number assignments by utilizing a curated list of resources and tools. This comprehensive guide directs students to valuable online platforms, textbooks, and interactive simulations. These resources enhance learning, offering diverse perspectives and aiding in the practical application of prime number concepts. As students navigate their mathematical journey, these tools become invaluable companions, facilitating a deeper grasp of prime numbers and ensuring a well-rounded approach to problem-solving.

In conclusion, "Prime Numbers, Prime Help" serves as a comprehensive handbook for students seeking excellence in math assignments centered around prime numbers. From historical perspectives to real-world applications, this guide offers a holistic understanding of prime numbers, providing a solid foundation for academic success and a deeper appreciation for the beauty of mathematics. As students navigate the intricate world of prime numbers, armed with knowledge and problem-solving skills, they are bound to excel in their mathematical endeavors.

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8.2: Prime Numbers and Prime Factorizations

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Preview Activity \(\PageIndex{1}\): Exploring Examples where \(a\) Divides \(b \cdot c\)

• Find at least three different examples of nonzero integers \(a\), \(b\), and \(c\) such that \(a\ |\ (bc)\) but a does not divide \(b\) and \(a\) does not divide \(c\). In each case, compute gcd(\(a\), \(b\)) and gcd(\(a\), \(c\)).
• Find at least three different examples of nonzero integers \(a\), \(b\), and \(c\) such that gcd(\(a\), \(b\)) = 1 and \(a\ |\ (bc)\). In each example, is there any relation between the integers \(a\) and \(c\)?
• Formulate a conjecture based on your work in Parts (1) and (2).

Preview Activity \(\PageIndex{2}\): Prime Factorizations

Recall that a natural number \(p\) is a prime number provided that it is greater than 1 and the only natural numbers that divide \(p\) are 1 and \(p\). A natural number other than 1 that is not a prime number is a composite number . The number 1 is neither prime nor composite. (See Exercise 13 from Section 2.4 on page 78.)

• Give examples of four natural numbers that are prime and four natural numbers that are composite.

Theorem 4.9 in Section 4.2 states that every natural number greater than 1 is either a prime number or a product of prime numbers. When a composite number is written as a product of prime numbers, we say that we have obtained a prime factorization of that composite number. For example, since \(60 = 2^2 \cdot 3 \cdot 5\), we say that \(2^2 \cdot 3 \cdot 5\) is a prime factorization of 60.

• Write the number 40 as a product of prime numbers by first writing \(40 = 2 \cdot 20\) and then factoring 20 into a product of primes. Next, write the number 40 as a product of prime numbers by first writing \(40 = 5 \cdot 8\) and then factoring 8 into a product of primes.
• In Part (2), we used two different methods to obtain a prime factorization of 40. Did these methods produce the same prime factorization or different prime factorizations? Explain.
• Repeat Parts (2) and (3) with 150. First, start with \(150 = 3 \cdot 50\), and then start with \(150 = 5 \cdot 30\).

Greatest Common Divisors and Linear Combinations

In Section 8.1 , we introduced the concept of the greatest common divisor of two integers. We showed how the Euclidean Algorithm can be used to find the greatest common divisor of two integers, \(a\) and \(b\), and also showed how to use the results of the Euclidean Algorithm to write the greatest common divisor of \(a\) and \(b\) as a linear combination of \(a\) and \(b\).

In this section, we will use these results to help prove the so-called Fundamental Theorem of Arithmetic, which states that any natural number greater than 1 that is not prime can be written as product of primes in “essentially” only one way. This means that given two prime factorizations, the prime factors are exactly the same, and the only difference may be in the order in which the prime factors are written. We start with more results concerning greatest common divisors. We first prove Proposition 5.16, which was part of Exercise (18) in Section 5.2 and Exercise (8) in Section 8.1.

Theorem 5.16

Let \(a\), \(b\), and \(t\) be integers with \(t \ne 0\). If \(t\) divides \(a\) and \(t\) divides \(b\), then for all integers \(x\) and \(y\), \(t\) divides \((ax + by)\).

Let \(a\), \(b\), and \(t\) be integers with \(t \ne 0\), and assuem that \(t\) divides \(a\) and \(t\) divides \(b\). We will prove that for all integers \(x\) and \(y\), \(t\) divides \((ax + by)\).

So let \(x \in \mathbb{Z}\) and let \(y \in \mathbb{Z}\). Since \(t\) divides \(a\), there exists an integer \(m\) such that \(a = mt\) and since \(t\) divides \(b\), there exists an integer \(n\) such that \(b = nt\). Using substitution and algebra, we then see that

\[\begin{array} {rcl} {ax + by} &= & {(mt) x + (nt) y} \\ {} &= & {t(mx + ny)} \end{array}\]

Since \((mx + ny\)) is an integer, the last equation proves that \(t\) divides \(ax + by\) and this proves that for all integers \(x\) and \(y\), \(t\) divides \((ax + by)\).

We now let \(a, b \in \mathbb{Z}\), not both 0, and let \(d = \text{gcd}(a, b)\). Theorem 8.8 states that d can be written as a linear combination of \(a\) and \(b\). Now, since \(d\ |\ a\) and \(d\ |\ b\), we can use the result of Proposition 5.16 to conclude that for all \(x, y \in \mathbb{Z}\), \(d\ |\ (ax + by)\). This means that \(d\) divides every linear combination of \(a\) and \(b\). In addition, this means that \(d\) must be the smallest positive number that is a linear combination of \(a\) and \(b\). We summarize these results in Theorem 8.9.

Theorem 8.9.

Let \(a, b \in \mathbb{Z}\), not both 0.

• The greatest common divisor, \(d\), is a linear combination of \(a\) and \(b\). That is, there exist integers \(m\) and \(n\) such that \(d = am + bn\).
• The greatest common divisor, \(d\), divides every linear combination of \(a\) and \(b\). That is, for all \(x, y \in \mathbb{Z}\), \(d\ |\ (ax + by)\).
• The greatest common divisor, \(d\), is the smallest positive number that is a linear combination of \(a\) and \(b\).

Relatively Prime Integers

In Preview Activity \(\PageIndex{1}\), we constructed several examples of integers \(a\), \(b\), and \(c\) such that \(a\ |\ (bc)\) but \(a\) does not divide \(b\) and \(a\) does not divide \(c\). For each example, we observed that \(\text{gcd}(a, b) \ne 1\) and \(\text{gcd}(a, c) \ne 1\).

We also constructed several examples where \(a\ |\ (bc)\) and \(\text{gcd}(a, b) = 1\). In all of these cases, we noted that \(a\) divides \(c\). Integers whose greatest common divisor is equal to 1 are given a special name.

Definition: relatively prime

Two nonzero integers \(a\) and \(b\) are relatively prime provided that \(\text{gcd}(a, b) = 1\).

Progress Check 8.10: Relatively Prime Integers

• Construct at least three different examples where \(p\) is a prime number, \(a \in \mathbb{Z}\), and \(p\ |\ a\). In each example, what is gcd(\(a\), \(p\))? Based on these examples, formulate a conjecture about gcd(\(a\), \(p\)) when \(p\ |\ a\).
• Construct at least three different examples where \(p\) is a prime number, \(a \in \mathbb{Z}\), and \(p\) does not divide \(a\). In each example, what is gcd(\(a, p\))? Based on these examples, formulate a conjecture about gcd(\(a\), \(p\)) when \(p\) does not divide \(a\).
• Give at least three different examples of integers \(a\) and \(b\) where a is not prime, \(b\) is not prime, and \(\text{gcd}(a, b) = 1\), or explain why it is not possible to construct such examples.

Add texts here. Do not delete this text first.

Theorem 8.11.

Let \(a\) and \(b\) be nonzero integers, and let \(p\) be a prime number.

• If \(a\) and \(b\) are relatively prime, then there exist integers \(m\) and \(n\) such that \(am + bn = 1\). That is, 1 can be written as linear combination of \(a\) of \(b\).
• If \(p\ |\ a\), then \(\text{gcd}(a, p) = p\).
• If \(p\) does not divide \(a\), then \(\text{gcd}(a, p) = 1\).

Part (1) of Theorem 8.11 is actually a corollary of Theorem 8.9. Parts (2) and (3) could have been the conjectures you formulated in Progress Check 8.10. The proofs are included in Exercise (1).

Given nonzero integers a and b, we have seen that it is possible to use the Euclidean Algorithm to write their greatest common divisor as a linear combination of \(a\) and \(b\). We have also seen that this can sometimes be a tedious, time-consuming process, which is why people have programmed computers to do this. Fortunately, in many proofs of number theory results, we do not actually have to construct this linear combination since simply knowing that it exists can be useful in proving results. This will be illustrated in the proof of Theorem 8.12, which is based on work in Preview Activity \(\PageIndex{1}\).

Theorem 8.12

Let \(a\), \(b\), be nonzero integers and let \(c\) be an integer. If \(a\) and \(b\) are relatively prime and \(a\ |\ (bc)\), then \(a\ |\ c\)

The explorations in Preview Activity \(\PageIndex{1}\) were related to this theorem. We will first explore the forward-backward process for the proof. The goal is to prove that \(a\ |\ c\). A standard way to do this is to prove that there exists an integer \(q\) such that

\[c = aq.\]

Since we are given \(a\ |\ (bc)\), there exists an integer \(k\) such that

\[bc = ak.\]

It may seem tempting to divide both sides of Equation \ref{8.2.3} by \(b\), but if we do so, we run into problems with the fact that the integers are not closed under division. Instead, we look at the other part of the hypothesis, which is that \(a\) and \(b\) are relatively prime. This means that \(\text{gcd}(a, b) = 1\). How can we use this? This means that \(a\) and \(b\) have no common factors except for 1. In light of Equation \ref{8.2.3}, it seems reasonable that any factor of \(a\) must also be a factor of \(c\). But how do we formalize this?

One conclusion that we can use is that since \(\text{gcd}(a, b) = 1\), by Theorem 8.11, there exist integers \(m\) and \(n\) such that

\[am + bn = 1.\]

We may consider solving equation (8.2.4) for \(b\) and substituting this into Equation \ref{8.2.3}. The problem, again, is that in order to solve Equation \ref{8.2.4} for \(b\), we need to divide by \(n\).

Before doing anything else, we should look at the goal in Equation \ref{8.2.2}. We need to introduce c into Equation \ref{8.2.4}. One way to do this is to multiply both sides of equation (8.2.4) by \(c\). (This keeps us in the system of integers since the integers are closed under multiplication.) This gives

\[\begin{array} {rcl} {(am + bn) c} &= & {1 \cdot c} \\ {acm + bcn} &= & {c.} \end{array}\]

Notice that the left side of Equation \ref{8.2.5} contains a term, \(bcn\), that contains \(bc\). This means that we can use Equation \ref{8.2.3} and substitute bc D ak in Equation \ref{8.2.5}. After doing this, we can factor the left side of the equation to prove that \(a\ |\ c\).

Progress Check 8.13: Completing the Proof of Theorem 8.12

Write a complete proof of Theorem 8.12.

Corollary 8.14

• Let \(a, b \in \mathbb{Z}\), and let \(p\) be a prime number. If \(p\ |\ (ab)\), then \(p\ |\ a\) or \(p\ |\ b\).
• Let \(p\) be a prime number, let \(n \in \mathbb{N}\), and let \(a_1, a_2, ..., a_n \in \mathbb{Z}\). If \(p\ |\ (a_{1}a_{2}\cdot\cdot\cdot a_{n})\), then there exists a natural number \(k\) with \(1 \le k \le n\) such that \(p\ |\ a_{k}\).

Part (1) of Corollary 8.14 is a corollary of Theorem 8.12. Part (2) is proved using mathematical induction. The basis step is the case where \(n = 1\), and Part (1) is the case where \(n = 2\). The proofs of these two results are included in Exercises (2) and (3).

Historical Note: Euclid’s Lemma

Part (1) of Corollary 8.14 is known as Euclid’s Lemma . Most people associate geometry with Euclid’s Elements, but these books also contain many basic results in number theory. Many of the results that are contained in this section appeared in Euclid’s Elements .

Prime Numbers and Prime Factorizations

We are now ready to prove the Fundamental Theorem of Arithmetic. The first part of this theorem was proved in Theorem 4.9 in Section 4.2. This theorem states that each natural number greater than 1 is either a prime number or is a product of prime numbers. Before we state the Fundamental Theorem of Arithmetic, we will discuss some notational conventions that will help us with the proof. We start with an example.

We will use \(n = 120\). Since \(5\ |\ 120\), we can write \(120 = 5 \cdot 24\). In addition, we can factor 24 as \(24 = 2 \cdot 2 \cdot 2 \cdot 3\). So we can write

\[\begin{array} {rcl} {120} &= & {5 \cdot 24} \\ {} &= & {5(2 \cdot 2 \cdot 2 \cdot 3).} \end{array}\]

This is a prime factorization of 120, but it is not the way we usually write this factorization. Most often, we will write the prime number factors in ascending order. So we write

\(120 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5\) or \(120 = 2^{3} \cdot 3 \cdot 5\).

Now, let \(n \in \mathbb{N}\). To write the prime factorization of \(n\) with the prime factors in ascending order requires that if we write \(n = p_{1}p_{2}\cdot\cdot\cdot p_{r}\), where \(p_{1}p_{2}\cdot\cdot\cdot p_{r}\) are prime numbers, we will have \(p_{1} \le p_{2} \le \cdot\cdot\cdot \le p_{r}\).

Theorem 8.15: The Fundamental Theorem of Arithmetic

• Each natural number greater than 1 is either a prime number or is a product of prime numbers.
• let \(n \in \mathbb{N}\) with \(n > 1\). Assume that \[n = p_{1}p_{2}\cdot\cdot\cdot p_{r} \text{ and that } n = q_{1}q_{2}\cdot\cdot\cdot q_{s},\] where \(p_{1}p_{2}\cdot\cdot\cdot p_{r}\) and \(q_{1}q_{2}\cdot\cdot\cdot q_{s}\) are prime with \(p_{1} \le p_{2} \le \cdot\cdot\cdot \le p_{r}\) and \(q_{1} \le q_{2} \le \cdot\cdot\cdot \le q_{s}\). Then \(r = s\), and for each \(j\) from 1 to \(r\), \(p_{j} = q{j}\).

The first part of this theorem was proved in Theorem 4.9. We will prove the second part of the theorem by induction on \(n\) using the Second Principle of Mathematical Induction. (See Section 4.2.) For each natural number \(n\) with \(n > 1\), let \(P(n)\) be

If \(n = p_{1}p_{2}\cdot\cdot\cdot p_{r}\) and \(n = q_{1}q_{2}\cdot\cdot\cdot q_{s}\), where \(p_{1}p_{2}\cdot\cdot\cdot p_{r}\) and \(q_{1}q_{2}\cdot\cdot\cdot q_{s}\) are primes with \(p_{1} \le p_{2} \le \cdot\cdot\cdot \le p_{r}\) and \(q_{1} \le q_{2} \le \cdot\cdot\cdot \le q_{s}\), then \(r = s\), and for each \(j\) from 1 to \(r\), \(p_{j} = q{j}\).

For the basis step, we notice that since 2 is a prime number, its only factorization is \(2 = 1 \cdot 2\). This means that the only equation of the form \(n = p_{1}p_{2}\cdot\cdot\cdot p_{r}\), where \(p_{1}p_{2}\cdot\cdot\cdot p_{r}\) are prime numbers, is the case where \(r = 1\) and \(p_1 = 2\).This proves that \(P(2)\) is true.

For the inductive step, let \(k \in \mathbb{N}\) with \(k \ge 2\). We will assume that \(P(2), P(3), ..., P(k)\) are true. The goal now is to prove that \(P(k + 1)\) is true. To prove this, we assume that \((k + 1)\) has two prime factorizations and then prove that these prime factorizations are the same. So we assume that

\(k + 1 = p_{1}p_{2}\cdot\cdot\cdot p_{r}\) and that \(k + 1 = q_{1}q_{2}\cdot\cdot\cdot q_{s}\), wher \(p_{1}p_{2}\cdot\cdot\cdot p_{r}\) and \(q_{1}q_{2}\cdot\cdot\cdot q_{s}\) are prime with \(p_{1} \le p_{2} \le \cdot\cdot\cdot \le p_{r}\) and \(q_{1} \le q_{2} \le \cdot\cdot\cdot \le q_{s}\).

We must now prove that \(r = s\), and for each \(j\) from 1 to \(r\), \(p_{j} = q_{j}\). We can break our proof into two cases: (1) \(p_{1} \le q_{1}\); and (2) \(q_{1} \le p_{1}\). Since one of these must be true, and since the proofs will be similar, we can assume, without loss of generality, that \(p_{1} \le q_{1}\).

Since \(k + 1 = p_{1}p_{2}\cdot\cdot\cdot p_{r}\), we know that \(p_{1}\ |\ (k + 1)\), and hence we may conclude that \(p_{1}\ |\ (q_{1}q_{2}\cdot\cdot\cdot q_{s})\). We now use Corollary 8.14 to conclude that there exists a \(j\) with \(1 \le j \le s\) such that \(p_{1}\ |\ q_{j}\). Since \(p_{1}\) and \(q_{j}\) are primes, we conclude that

\(p_{1} = q_{j}\).

We now use this and the fact that \(k + 1 = p_{1}p_{2}\cdot\cdot\cdot p_{r} = q_{1}q_{2}\cdot\cdot\cdot q_{s}\) to conclude that

\(p_{2}\cdot\cdot\cdot p_{r} = q_{2}\cdot\cdot\cdot q_{s}\).

The product in the previous equation is less that \(k + 1\). Hence, we can apply our induction hypothesis to these factorizations and conclude that \(r = s\), and for each \(j\) from 2 to \(r\), \(p_{j} = q_{j}\).

This completes the proof that if \(P(2), P(3), ..., P(k)\) are true, then \(P(k + 1)\) is true. Hence, by the Second Principle of Mathematical Induction, we conclude that \(P(n)\) is true for all \(n \in \mathbb{N}\) with \(n \ge 2\). This completes the proof of the theorem.

Note : We often shorten the result of the Fundamental Theorem of Arithmetic by simply saying that each natural number greater than one that is not a prime has a unique factorization as a product of primes. This simply means that if \(n \in \mathbb{N}\), \(n > 1\), and n is not prime, then no matter how we choose to factor n into a product of primes, we will always have the same prime factors. The only difference may be in the order in which we write the prime factors.

Further Results and Conjectures about Prime Numbers

Theorem 8.16..

There are infinitely many prime numbers.

We will use a proof by contradiction. We assume that there are only finitely many primes, and let

\(p_{1}, p_{2}, ..., p_{m}\)

be the list of all the primes. Let

\[M = p_{1}p_{2}, ..., p_{m} + 1.\]

Notice that \(M \ne 1\). So \(M\) is either a prime number or, by the Fundamental Theorem of Arithmetic, \(M\) is a product of prime numbers. In either case, \(M\) has a factor that is a prime number. Since we have listed all the prime numbers, this means that there exists a natural number \(j\) with \(1 \le j \le m\) such that \(p_{j}\ |\ M\). Now, we can rewrite equation (8.2.8) as follows:

\[1 = M - p_{1}p_{2} \cdot\cdot\cdot p_{m}.\]

We have proved \(p_{j}\ |\ M\), and since \(p_{j}\) is one of the prime factors of \(p_{1}p_{2} \cdot\cdot\cdot p_{m}\), we can also conclude that \(p_{j}\ |\ (p_{1}p_{2}\cdot\cdot\cdot p_{m})\). Since \(p_{j}\) divides both of the terms on the right side of equation (8.2.9), we can use this equation to conclude that \(p_{j}\) divides 1. This is a contradiction since a prime number is greater than 1 and cannot divide 1. Hence, our assumption that there are only finitely many primes is false, and so there must be infinitely many primes.

Theorem 8.17,

For any natural number \(n\), there exist at least \(n\) consecutive natural numbers that are composite numbers.

There are many unanswered questions about prime numbers, two of which will now be discussed.

For some interesting information on prime numbers, visit the Web site The Prime Pages (primes.utm.edu/), where there is a link to The Largest Known Primes Web site. According to information at this site as of June 25, 2010, the largest known twin primes are \[(65516468355 \times 2^{333333} - 1) \text{ and } (65516468355 \times 2^{333333} + 1).\] Each of these prime numbers contains 100355 digits.

One of the most famous unsolved problems in mathematics is a conjecture made by Christian Goldbach in a letter to Leonhard Euler in 1742. The conjecture, now known as Goldbach’s Conjecture , is as follows:

Every even integer greater than 2 can be expressed as the sum of two (not necessarily distinct) prime numbers.

As of June 25, 2010, it is not known if this conjecture is true or false, al- though most mathematicians believe it to be true.

Exercise 8.2

• Prove the second and third parts of Theorem 8.11. (a) Let \(a\) be a nonzero integer, and let \(p\) be a prime number. If \(p\ |\ a\), then \(\text{gcd}(a, p) = p\). (b) Let \(a\) be a nonzero integer, and let \(p\) be a prime number. If \(p\) does not divide \(a\), then \(\text{gcd}(a, p) = 1\).
• Prove the first part of Corollary 8.14. Let \(a, b \in \mathbb{Z}\), let \(p\) be a prime number. If \(p\ |\ (ab)\), then \(p\ |\ a\) or \(p\ |\ b\). Hint : Consider two cases: (1) \(p\ |\ a\); and (2) \(p\) does not divide \(a\).
• Use mathematical induction to prove the second part of Corollary 8.14. Let \(p\) be a prime number, let \(n \in \mathbb{N}\), and let \(a_1, a_2, ..., a_n \in \mathbb{Z}\). If \(p\ |\ (a_{1}a_{2} \cdot\cdot\cdot a_{n})\), then there exists a \(k \in \mathbb{N}\) with \(1 \le k \le n\) such that \(p\ |\ a_k\).
• (a) Let \(a\) and \(b\) be nonzero integers. If there exist integers \(x\) and \(y\) such that \(ax + by = 1\), what conclusion can be made about gcd(\(a, b\))? Explain. (b) Let \(a\) and \(b\) be nonzero integers. If there exist integers \(x\) and \(y\) such that \(ax + by = 2\), what conclusion can be made about gcd(\(a, b\))? Explain.

Hint : Exercise (4) might be helpful.

(b) Let \(a \in \mathbb{Z}\). What conclusion can be made about gcd(\(a\), \(a + 2\))? That is, what conclusion can be made about the greatest common divisor of two integers that differ by 2? Justify your conclusion.

• (a) Let \(a \in \mathbb{Z}\). What conclusion can be made about gcd(\(a\), \(a + 3\))? That is, what conclusion can be made about the greatest common divisor of two integers that differ by 3? Justify your conclusion. (b) Let \(a \in \mathbb{Z}\). What conclusion can be made about gcd(\(a\), \(a + 4\))? That is, what conclusion can be made about the greatest common divisor of two integers that differ by 4? Justify your conclusion.
• (a) Let \(a = 16\) and \(b = 28\). Determine the value of \(d = \text{gcd}(a, b)\), and then determine the value of gcd(\(\dfrac{a}{d}\), \(\dfrac{b}{d}\)). (b) Repeat Exercise (7a) with \(a = 10\) and \(b = 45\). (c) Let \(a, b \in \mathbb{Z}\), not both equal to 0, and let \(d = \text{gcd}(a, b)\). Explain why \(\dfrac{a}{d}\) and \(\dfrac{b}{d}\) are integers. Then prove that \(\text{gcd}(\dfrac{a}{d}, \dfrac{b}{d}) = 1\). Hint: Start by writing \(d\) as a linear combination of \(a\) and \(b\). This says that if you divide both \(a\) and \(b\) by their greatest common divisor, the result will be two relatively prime integers.
• Are the following propositions true or false? Justify your conclusions. (a) For all integers \(a\), \(b\), and \(c\), if \(a\ |\ c\) and \(b\ |\ c\), then \((ab)\ |\ c\). (b) For all integers \(a\), \(b\), and \(c\), if \(a\ |\ c\), \(b\ |\ c\), and \(\text{gcd}(a, b) = 1\), then \((ab)\ |\ c\).
• In Exercise (16) in Section 3.5, it was proved that if \(n\) is an odd integer, then \(8\ |\ (n^2 - 1\)\). (This result was also proved in Exercise (19) in Section 7.4.) Now, prove the following proposition: If \(n\) is an odd integer and 3 does not divide \(n\), then \(24\ |\ (n^2 - 1)\).
• (a) Prove the following proposition: For all \(a, b, c \in \mathbb{Z}\), \(\text{gcd}(a, bc) = 1\) if and only if \(\text{gcd}(a, b) = 1\) and \(\text{gcd}(a, c) = 1\). (b) Use mathematical induction to prove the following proposition: Let \(n \in \mathbb{N}\) and let \(a, b_1, b_2, ..., b_n \in \mathbb{Z}\). If \(\text{gcd}(a, b_i) = 1\) for all \(i \in \mathbb{N}\) with \(1 \le i \le n\), then \(\text{gcd}(a, b_{1}b_{2} \cdot\cdot\cdot b_{n}) = 1\).
• Is the following proposition true or false? Justify your conclusion. Fro all integer \(a\), \(b\), and \(c\), if \(\text{gcd}(a, b) = 1\) and \(c\ |\ (a + b)\), then \(\text{gcd}(a, c) = 1\) and \(\text{gcd}(b, c) = 1\).
• Is the following proposition true or false? Justify your conclusion. If \(n \in \mathbb{N}\), then \(\text{gcd}(5n + 2, 12n + 5) = 1\)
• Let \(y \in \mathbb{N}\). Use the Fundamental Theorem of Arithmetic to prove that there exists an odd natural number x and a nonnegative integer \(k\) such that \(y = 2^{k}x\).
• (a) Determine five different primes that are congruent to 3 modulo 4. (b) Prove that there are infinitely many primes that are congruent to 3 modulo 4.
• (a) Let \(n \in \mathbb{N}\). Prove that 2 divides \([(n + 1)! + 2]\). (b) Let \(n \in \mathbb{N}\) with \(n \ge 2\). Prove that 3 divides \([(n + 1)! + 3]\). (c) Let \(n \in \mathbb{N}\). Prove that for each \(k \in \mathbb{N}\) with \(2 \le k \le (n + 1)\), \(k\) divides \([(n + 1)! + k]\). (d) Use the result of Exercise (15c) to prove that for each \(n \in \mathbb{N}\), there exist at least \(n\) consecutive composite natural numbers.
• The Twin Prime Conjecture states that there are infinitely many twin primes, but it is not known if this conjecture is true or false. The answers to the following questions, however, can be determined. (a) How many pairs of primes \(p\) and \(q\) exist where \(q - p = 3\)? That is, how many pairs of primes are there that differ by 3? Prove that your answer is correct. (One such pair is 2 and 5.) (b) How many triplets of primes of the form \(p\), \(p + 2\), and \(p + 4\) are there? That is, how many triplets of primes exist where each prime is 2 more than the preceding prime? Prove that your answer is correct. Notice that one such triplet is 3, 5, and 7. Hint : Try setting up cases using congruence modulo 3.
• Prove the following proposition: Let \(n \in \mathbb{N}\). For each \(a \in \mathbb{Z}\), if \(\text{gcd}(a, n) = 1\), then for every \(b \in \mathbb{Z}\), there exists an \(x \in \mathbb{Z}\) such that \(ax \equiv b\) (mod \(n\)). Hint: One way is to start by writing 1 as a linear combination of \(a\) and \(n\).

For all natural numbers \(m\) and \(n\), if \(m\) and \(n\) are twin primes other than the pair 3 and 5, then 36 divides \(mn + 1\) and \(mn + 1\) is a perfect square.

Hint : Look at several examples of twin primes. What do you notice about the number that is between the two twin primes? Set up cases based on this observation.

Explorations and Activities

19. Square Roots and Irrational Numbers. In Chapter 3, we proved that some square roots (such as \(\sqrt{2}\) and \(\sqrt{3}\)) are irrational numbers. In this activity, we will use the Fundamental Theorem of Arithmetic to prove that if a natural number is not a perfect square, then its square root is an irrational number. (a) Let \(n\) be a natural number. Use the Fundamental Theorem of Arithmetic to explain why if n is composite, then there exist prime numbers \(p_{1}, p_{2}, ..., p_{r}\) and natural numbers \(\alpha_{1}, \alpha_{2}, ..., \alpha_{r}\) such that

\[n = p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}} \cdot\cdot\cdot p_{r}^{\alpha_{r}}.\]

Then,if we use \(r = 1\) and \(\alpha_{1} = 1\) for a prime number, explain why we can write any natural number in the form given in equation (8.2.11).

(b) A natural number \(b\) is a perfect square if and only if there exists a natural number \(a\) such that \(b = a^2\). Explain why 36, 400, and 15876 are perfect squares. Then determine the prime factorization of these perfect squares. What do you notice about these prime factorizations?

(c) Let \(n\) be a natural number written in the form given in equation (8.2.11) in part (a). Prove that \(n\) is a perfect square if and only if for each natural number \(k\) with \(1 \le k \le r\), \(\alpha_{k}\) is even.

(d) Prove that for all natural numbers \(n\), if \(n\) is not a perfect square, then \(\sqrt{n}\) is an irrational number. Hint : Use a proof by contradiction.

Mathematics Doubts

Prime number.

• Active page

Sep 06, 2022, 16:45 IST

All natural numbers that have one and itself only as their factors are called prime numbers i.e. prime numbers are exactly divisible by 1 and themselves. For More Mathematics Doubts visit main page of Physcis Wallah.

e.g. 2,3,5,7,11,13,17,19,23....etc.

If P is the set of prime number then P = {2,3,5,7....}.

List of Prime numbers

The list of all prime numbers from 1 to 100 is given below:

Properties of Prime Numbers

• Every number greater than 1 is divisible by at least 1 prime number.
• Every even positive integer which is greater than 2 can be expressed as the sum of two prime numbers.
• Except for 2, all other prime numbers are odd. As, we can say that 2 is the only even prime number.
• Two prime numbers are always coprime.
• Every composite number can be divided into prime factors; individually, they are all unique.

Methods to find Prime Numbers

We all know that 2 is the only even prime number. And the only two consecutive natural numbers that are prime are 2 and 3. Apart from them, every prime number can be written as 6n + 1 or 6n – 1 (except for multiples of primes, i.e., 2, 3, 5, 7, 11), where n is a natural number.

For example: 6(1) – 1 = 5

6(1) + 1 = 7

6(2) – 1 = 11

6(2) + 1 = 13

6(3) – 1 = 17

6(3) + 1 = 19

The formula below can be used to know the prime numbers greater than 40.

n 2 + n + 41, where n = 0, 1, 2, ….., 39

For example:

(0) 2 + 0 + 0 = 41

(1) 2 + 1 + 41 = 43

(2) 2 + 2 + 41 = 47

List of Odd Prime Numbers

A prime number chart is a chart that displays a list of prime numbers in standard order. It is important to know that all prime numbers are odd numbers except for the number 2, which is the only even number. Therefore, 2 is the only even prime number. This means that the list of odd numbers can start from 3 and continue since the remaining prime numbers are odd. For eg, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and so on are odd prime numbers.

History of Prime Numbers

A Prime number was discovered by Eratosthenes (275-194 BC, Greece). He took the example of a sieve to separate the prime numbers from the list of natural numbers and exhaust the composite numbers. Students can practice this method by writing down the positive integers from 1 to 100, circling the primes, and crossing the composites. This kind of all activity refers to the Sieve of Eratosthenes.

Solved Examples

Q1. Is 23 a prime number?

Ans. Yes, it is a prime number. The factor of 23 are 1 and 23

Lets, write the given number in the form of 6n-1

6 (4) - 1 = 23.

Q2. Is 19 a Prime number?

Ans. Yes, it is a prime number. The factor of 19 are 1 and 19

Lets, write the given number in the form of 6n + 1

6 (3) + 1 = 19.

Q3. Why 20 is not a prime number?

Ans. 1, 2, 4, 5, 10, and 20 are the factor of 20. Therefore, 20 has more than two factors. Since the factors of 20 is more than two numbers, it is NOT a prime number (20 is a composite number).

Frequently Asked Question (FAQs)

Q1. Is 0 a prime number?

Ans. No, zero is not a prime number.

Q2. What is the next prime number after 7?

Ans. The next prime number after 7 is 11,13,17,19,23,29 and so on.

Q3. Can negative numbers be prime?

Ans. No, Negative integers cannot be prime by the standard definition of prime for integers. By this definition, primes are integers greater than one with no positive divisors besides one and itself. Negative numbers are excluded.

Q4. How do you know if a number is prime or not?

Ans. To know whether a number is a prime number, try dividing it by two and then see if you can get a whole number. It can't be a prime number if you get a whole number. If you don't get a whole number, try dividing it by prime numbers, i.e., 3, 5, 7, 11, and soon.

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Prime Numbers

General purpose of this lecture is to present on Prime Numbers. A natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it has exactly two positive divisors, 1 and the number itself. Even Numbers – Any number that can be divided by 2. Odd Numbers – Any number that cannot be divided by 2. Composite Number – An integer that can be divided by at least one other number (a factor) other than itself. Prime Number – An integer whose only factors are 1 and itself. Factor – a number that can divide another number without a remainder.

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Prime Number Puzzle Has Stumped Mathematicians for More Than a Century

Experts have only started to crack the tricky twin prime conjecture

By Manon Bischoff

FrankRamspott/Getty Images

While I was looking for a gift for a child’s birthday, a math book fell into my hands . I am always fascinated when authors write about abstract scientific topics for children, whether it’s on Albert Einstein’s theories, the life of Marie Curie, technology or space travel. But this particular book was different. It’s all about prime numbers—specifically twin primes. Danish author Jan Egesborg has endeavored to introduce children to one of the most stubborn open problems in number theory, which even the brightest minds have repeatedly failed to solve over the past 100-plus years: the twin prime conjecture.

As is so often the case in mathematics, the conjecture falls into the category of those that are easy to understand but devilishly hard to prove. Twin primes are two prime numbers that have a distance of two on the number line; that is, they are directly consecutive if you ignore even numbers. Examples include 3 and 5, 5 and 7, and 17 and 19. You can find a lot of twin primes among small numbers, but the farther up the number line you go, the rarer they become.

That’s no surprise, given that prime numbers are increasingly rare among large numbers. Nevertheless, people have known since ancient times that infinite prime numbers exist, and the prime number twin conjecture states that there are an infinite number of prime number twins, as well. That would mean that no matter how large the values considered, there will always be prime numbers in direct succession among the odd numbers.

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Admittedly, translating these concepts for kids is not easy (which is why I have so much respect for Egesborg and his children’s book). Prime numbers (2, 3, 5, 7, 11, 13,...) are like the fundamental particles of the natural numbers. They are only divisible by 1 and themselves. All other natural numbers can be broken down into their prime divisors, which makes prime numbers the basic building blocks of the mathematical world.

A Proof from Antiquity

Mathematics has an unlimited number of prime number building blocks. Euclid proved this more than 2,000 years ago with a simple thought experiment. Suppose there were only a finite number of prime numbers, the largest being p . In this case, all prime numbers up to p could be multiplied together.

In this case, you could multiply all prime numbers up to p with each other and add 1: 2 x 3 x 5 x 7 x 11 x ... x p + 1. The result cannot be divided by any of the existing prime numbers. This means that the number 2 x 3 x 5 x 7 x 11 x ... x p + 1 is either prime or has a prime factor that does not appear in the original 2, 3,..., p primes. Therefore, no finite list of primes can ever be complete; it will always be possible to construct additional ones. It follows that there are infinite prime numbers.

Not all mysteries about prime numbers have been solved, however. Their distribution on the number line, in particular, remains a mystery. Although we know that prime numbers appear less and less frequently among large numbers, it is not possible to specify exactly how they are distributed.

In principle, the average distance between one prime number and the next is the value ln( p ). For the small number p = 19, this corresponds to ln(19) ≈ 3. For the large prime number 2,147,483,647, the distance is around 22. For the huge value 531,137,992,816,767,098,689,588,206,552,468,627,329,593,117,727,031,923,199,444,138,200,403,559,860,852,242,739,162,502,265,229,285,668,889,329,486,246,501,015,346,579,337,652,707,239,409,519,978,766,587,351,943,831,270,835,393,219,031,728,127 (also a prime number), the distance is around 420.

As these examples illustrate, the average distance between the prime numbers increases with the size of p . And this fact makes prime number twins, which have the smallest possible distance between them (apart from 2 and 3), so interesting to number theorists. As the average distance between prime numbers increases, it could be that at a certain point there are no more twins. Yet most experts think otherwise. Why, they reason, should there be a certain point on the number line from which no more twin primes suddenly appear? What makes this one point so special? Number theorists assume that even if these prime number twins become rarer, you will always eventually come across another pair.

Computer calculations to date seem to support this view. The largest pair of prime number twins found so far is: 2,996,863,034,895 x 2 1,290,000 + 1 and 2,996,863,034,895 x 2 1,290,000 – 1, both numbers with 388,342 digits. A computer-aided search will never be able to prove that there are an infinite number of twin primes, however. Stronger tactics are needed.

An Unexpected Surprise

A little-known mathematician delivered just that in 2013. Yitang Zhang had previously been a household name among very few specialists—but then he published a paper that hit the number theory world like a bomb. He was not able to prove the prime number twin conjecture but demonstrated something close to it, which was more progress than anyone had made since the twin prime conjecture was formulated in the 19th century.

Zhang showed that there are an infinite number of pairs of prime numbers of the type ( p , p + N ) with a distance N between them that is less than 70 million. The twin prime conjecture would have been proved if he had been able to prove his result for N = 2. Instead Zhang demonstrated that among all pairs of prime numbers with a distance of less than 70 million, there is at least one pairing ( p , p + N ) that occurs infinitely often.

This proof was a huge step forward because mathematicians are not only interested in prime number twins but also in other types of prime number pairs, such as those with a distance of four (such as 3 and 7 or 19 and 23), the so-called cousin primes, or those with a distance of six (such as 5 and 11 or 11 and 17), the so-called sexy primes. In general, it is unclear whether an infinite number of any of these pairings exist.

Zhang achieved this astonishing result using what mathematicians call prime number sieves. These constructs can be imagined as a real sieve: you tip all the natural numbers into it and filter out all the values that are not prime numbers. This idea is named for the ancient Greek scholar and mathematician Eratosthenes, though the first known written record of it is from a few centuries after he lived. It involves a list of natural numbers in which one removes every even value (apart from 2), then all multiples of 3, multiples of 5, and so on, such that only the prime numbers remain at the end.

By going through all the natural numbers one by one and eliminating their multiples (except for the number itself), only prime numbers will remain.

Amanda Montañez

Although the sieve of Eratosthenes is exact, it is very difficult to apply to concrete problems from a mathematical point of view. Using this method to prove general statements about prime numbers seems hopeless in most cases. Zhang therefore turned to another sieve that only sifts out numbers with large prime divisors. Although this sieve is not as effective as others, it allows enough flexibility to carry out extensive proofs. Zhang worked single-handedly on the twin prime conjecture for years—number theory was not actually part of his research area.

This persistence paid off: Zhang proved that there is at least one kind of prime number pair with a distance of less than 70 million that occurs infinitely often. And the next breakthrough was not long in coming.

Number theorists from all over the world pounced on Zhang’s result and tried to improve it. A joint project was set up, and numerous experts joined in. By optimizing Zhang’s method, they were able to reduce the maximum distance N between the pairs of prime numbers to get as close as possible to 2. Within a few months , they showed that there is at least one type of prime number pair with a maximum distance of 4,680 that occurs infinitely often. Around the same time, two Fields Medalists, Terence Tao and James Maynard, independently developed a modified sieve that enabled them to reduce the result to 246, an unbroken record to date.

In concrete terms, this means that if you look at all pairs of prime numbers ( p , p + N ) that have a distance between N = 2 and N = 246, then there is at least one such pair that occurs infinitely often. The sieving methods cannot be generalized so far as to push the result down to N = 2, however.

Still, the results mark unexpected progress in an area that leaves many experts baffled. Maynard makes this clear in a Numberphile YouTube video: “This is one of the interesting and frustrating things about prime numbers: that often it’s clear what the right answer should be.... The game is always trying to rule out there being some very bizarre conspiracy among prime numbers that would mean that they would behave in a rather different way to how we believe that they should behave.”

Of course, Egesborg could not include all these details in his children’s book on the subject. Nevertheless, he managed to write a book that conveys a few mathematical concepts in a playful way.

I bought the book and gave it to the child in question on his birthday—and. his parents later told me that he had thoroughly enjoyed it. As I found out afterward, however, this was less a result of the mathematical content than the fact that a frog farts loudly on one of the first pages.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

Mathematical definitions are precise, each word chosen with care.  A prime example is below:

Prime Number: A number that has exactly two factors.

Composite Numbers have more than two factors.

The first prime number is two.  The factors of two are 1 and 2.  That makes two total factors, making two prime.  Since two divides into all even numbers (why they’re called even), two is the only even prime number.  An example would be 20.  Factors of twenty include 1 and 20, and 2 and 10.  That’s more than two factors, making twenty composite.

One way to determine if a number is prime or composite is to try and divide the number by primes smaller than it.  If three, for example, divides evenly into 4,695, then it is composite.  Note that since 4,695 ends in a five, five divides into evenly, so we already know it is composite.  But there’s a nifty trick, a short cut, to check divisibility by three. 

To see if a number is divisible by three, add the digits together.  If the sum is divisible by three, then so is the entire number.  Our number 4,695 for example adds to 24, which is divisible by 3.  Take 24, 2 + 4 = 6, which is also divisible by three.

Perhaps the easiest way to know if a number is prime or composite is to be familiar with the primes less than 100.  There are twenty-five and they are listed below:

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89, 97

What About One?

Since one has just one factor, itself, it does not satisfy the definition of prime or composite. 

This is not just a minor issue, one being neither prime or composite.  The Fundamental Theorem of Algebra says, basically, that every number can be uniquely expressed as the product of prime numbers.  For example, 14 is the product of 2 and 7.  If one was prime, then there would be another way to write 14 as the product of primes, 1, 2 and 7.  This has major consequences.  So the math seems to work best if one is not prime or composite.

Prime versus Relative Prime

Prime numbers have just two factors, and you only consider the number itself.  When discussing relatively prime numbers you are comparing more than one number to another. If they share more a factor other than one, they are not relatively prime.

Relatively prime numbers will share exactly one factor, the number one.

An example of a relatively prime pair of numbers would be 15 and 77.  Neither number is prime, but 15 has factors of 1 and 3 and 5 and 15.  Seventy-seven has factors of 1 and 7 (for this case we only need list one of the sevens), and 11.  The only common factor they share is one. 

An example of numbers that are not relatively prime is 15 and 9.  They share a factor of one and of three. 

If two numbers are prime are they also relatively prime?  How do you know?

Factoring Numbers

A factor is a part of a number.  Factors multiply together to make the number at hand.  For example, the factors of 24 are 1, 24 and 2, 12, and 3, 8 and 4, 6.  This would be a complete factorization of 24.

The prime factors of 24 are: 2, 2, 2, 3. 

The purpose of factoring is often to find some property of the number in question.  Sometimes you will factor to determine if a perfect square or perfect cube exists as a factor of a number. 

Other times you will be looking for a lowest common multiple so that you can find a common denominator, for something like:

What About Variables and Factoring?

In Algebra you will be factoring algebraic expressions like 15 x 2 .  The confusion comes in with how to factor the x 2 .  To factor x 2 we would just write x•x . We will discuss this in greater detail when we learn how to find the lowest common multiple and greatest common factors of algebraic expressions in the next section.

Perfect Squares and Perfect Cubes

A perfect square is the product of a number times itself.  Examples would include 1, 36,49, 400, x 2 and a 4 .  Each of these is the product of a number squared (times itself).

A perfect cube is the product of a number times itself three times (cubed).

There is an interesting pattern in perfect squares.  See it below?

How many prime numbers exist?  Take a guess.  How do you know?  In a future section we will introduce mathematical proof and discuss this question.

In summary, a prime number has exactly two factors, one and itself.  Composite numbers have more than two factors, and one is neither prime nor composite. 

Prime Numbers  Lesson Plan

This is a one or two-day lesson.  You can trim the PowerPoint to one day, or explore a second day to really encourage student thinking. 

Note:   The objectives of this teaching this topic include:

• Create mystery and interest in mathematics through questioning
• Promote mathematical thinking
• Introduce the idea that an algebraic expression could be prime
• Knowledge of prime, versus composite, and how the number 1 fits
• Knowledge of prime numbers less than 100

The simple definition of prime is not sufficient to achieve all of these objectives.  It will be important for you, the teacher, to be creative in questioning and forcing students to articulate their thinking.  This material will provide you some ideas. 

Numbers can be built, and deconstructed with multiplication.  This comes into play later with LCM and GCF, and with algebraic expressions, it becomes quite confusing because students mix up the meaning of multiple and factor.

Key Knowledge

• A prime number has exactly two factors, one and itself.
• A composite number has more than two factors.
• One is not prime or composite because it has exactly one factor.

Pro-Tip (for students)

Know the prime numbers less than 100, especially those numbers close to 100.  Understand that an Algebraic expression can be prime!

To download the PowerPoint, click the icon below.

Prime Numbers Practice Problems                                                                             

• The two values of n that are prime.
• Could n be a perfect square?
• Could n be a perfect cube?
• Why is one not composite or prime?
• Describe the short cut divisibility check for three.
• Is 5,821 prime or composite?
• The number a is divisible by 7. What could the last digit of the number a be?
• If Relative Prime Numbers are numbers that only share one common factor, the number one, are 12 and 39 relatively prime? How do you know?

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• Topic Reference Sheet
• Lesson Guide
• Quiz and Key (Prime and Real Numbers)

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Prime Numbers Chart and Calculator

A Prime Number is:

a whole number above 1 that cannot be made by multiplying other whole numbers

(if we can make it by multiplying other whole numbers it is a Composite Number )

Here we see it in action:

Here is a list of all the prime numbers up to 1,000:

• Is 8 a Prime Number? No , because it can be made by 2×4=8
• Is 73 a Prime Number? Yes , as no other whole numbers multiply together to make it.

Calculator ... Is It Prime?

Find out if a number is Prime or not (works on numbers up to 4,294,967,295):

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• Saved Saved

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Election Results 2024 Live: Congress wins 4 of 6 seats in Himachal assembly bypolls

Election Results 2024 Live: Four Congress turncoats, who recently joined the BJP in Himachal Pradesh, on Tuesday lost the by-elections from their respective assembly segments, the Election Commission figures showed.

With victory in four assembly segments, the strength of Congress MLAs has gone up to 38 in the house with an effective strength of 65. The result ensures that there is no immediate threat to the Congress government in the state for now.

Election Results 2024 Live: Rajnath Singh wins Lucknow

Election Results 2024 Live: BJP candidate and Defence Minister Rajnath Singh wins Lucknow Lok Sabha seat with a margin of 1,35,159 votes: Election Commission.

Election Results 2024 Live: Another BJP union minter defeated

Election Results 2024 Live: Union minister and BJP candidate Ajay Mishra Teni lost the Lakhimpur Kheri Lok Sabha seat to the Samajwadi Party’s Utkarsh Verma by a margin of 34,329 votes, according to Election Commission figures.

Election Results 2024 Live: Wayanad or Rae Bareli, which constituency will Rahul Gandhi choose?

Election Results 2024 Live: “Thank you very much to the people of Wayanad and Raebareli for giving me immense love. If it was in my control, I would have liked to remain an MP from both the places." tweets rahul Gandhi

Election Results 2024 Live: INDIA bloc will meet on Wednesday to decide on PM face

Election Results 2024 Live: Uddhav Thackeray says INDIA bloc will meet on Wednesday to decide on PM face. “Congress, other (leaders) are having parleys with them: Thackeray on whether Nitish Kumar, Chandrababu Naidu will join INDIA bloc.", said Uddhav Thackeray. 

Addressing reporters in Mumbai, Thackeray said, ‘Common man had shown his strength today. When the powerful become egoistic, people can bring them down. I will go to Delhi post afternoon tomorrow. Sanjay Raut will go for the meeting tomorrow.’

Election Results 2024 Live: Modi has no moral right to continue as PM, says Siddaramaiah

Election Results 2024 Live: Karnataka Chief Minister Siddaramaiah on Tuesday said Narendra Modi has no moral right to become prime minister again as the Lok Sabha polls verdict showed that there was no ‘Modi Wave’ anywhere and that his "popularity has diminished".

Election Results 2024 Live: Latest EC trends 

Election Results 2024 Live: Here are latest EC trends  Bharatiya Janata Party -143 (Won) 97 (Leading) 

Indian National Congress - 57 (Won) 42 (Leading)

Election Results 2024 Live: PM Modi arrives at BJP headquarters in Delhi

Election Results 2024 Live: After a muted Lok Saha win, PM Modi arrived at BJP headquarters in Delhi

Election Results 2024 Live: BJP's Ramvir Singh Bidhuri wins from South Delhi

Election Results 2024 Live: BJP's Ramvir Singh Bidhuri won the South Delhi seat, defeating AAP's Sahi Ram Pahalwan by a margin of 1,24,333 votes.

Bidhuri is the leader of opposition in the Delhi Assembly.

He was fielded by the BJP in place of outgoing MP Ramesh Bidhuri. The BJP won all seven seats in Delhi in the 2014 and the 2019 general elections, and is set to repeat the feat this time too.

In the 2024 Lok Sabha polls, the Congress fielded candidates in three of the seven Lok Sabha seats in Delhi and the Aam Aadmi Party in four as part of a seat-sharing agreement between the two constituents of the INDIA bloc.

Election Results 2024 Live: Union cabinet meeting likely to be held tomorrow, in Delhi.

Election Results 2024 Live: BJP set to lose majority

Election Results 2024 Live: Prime Minister Narendra Modi was the message and the messenger for the BJP in the Lok Sabha polls, dwarfing all his rivals in the gruelling campaign.

But as results poured in on Tuesday, the message seemed to have been overshadowed by the resurgent opposition, which restricted the right-wing party to far below the majority mark of 272 in the 543-member Lok Sabha. The BJP-led NDA, however, is expected to cross the magic figure comfortably.

Election Results 2024 Live: NDA performs strongly in metros while INDIA bloc dominates rural areas

Election Results 2024 Live: Election Results 2024 Live:The National Democratic Alliance (NDA) has shown significant strength in the metro cities in the Lok Sabha elections, securing substantial leads in urban centers such as Delhi, Bangalore and Pune.

However, Mumbai region defied the trend.

The INDIA bloc has demonstrated its prowess in the rural regions, capturing the majority of votes in the countryside.

Election Results 2024 Live: AAP bags three Lok Sabha seats in Punjab,

Election Results 2024 Live: The ruling Aam Aadmi Party in Punjab could manage to victory in just three Lok Sabha seats in Punjab, falling way short of Chief Minister Bhagwant Mann's stated target of winning all 13 parliamentary constituencies of the state.

On the other hand, the Congress has bagged six seats and is leading on one while the Shiromani Akali Dal won just one seat.

Election Results 2024 Live: PM Modi thanks AP, Odisha

Election Results 2024 Live: PM Modi thanked Andhra Pradesh and Odisha for the resounding victory in the Assembly polls 

Election Results 2024 Live: Ashok Gehlot says ‘this is the win of the people’

Election Results 2024 Live: Former Rajasthan CM & Congress leader Ashok Gehlot says, "This election became NDA vs the people...This is the win of the people...I had said earlier too that there will be shocking results from Rajasthan...I had belief in the people and party workers..."

Election Results 2024 Live: ‘This is hisotorical…’ says PM Modi

Election Results 2024 Live: “People have placed their faith in NDA, for a third consecutive time! This is a historical feat in India’s history. I bow to the Janata Janardan for this affection and assure them that we will continue the good work done in the last decade to keep fulfilling the aspirations of people. I also salute all our Karyakartas for their hard work. Words will never do justice to their exceptional efforts." PM Modi.

Election Results 2024 Live:'How's the josh?'- it’s still high, Sir' tweets Smriti Irani, conceding defeat

Election Results 2024 Live: BJP leader Smriti irani took to microblogging site X to say "Such is life... A decade of my life going from one village to another, building lives, nurturing hope & aspirations, working on infrastructure ― roads, naali, khadanja, bypass, medical college and more.

To those who stood by me through loss and victory, I am forever grateful. To those celebrating today, congratulations. And to those asking, "How's the josh?" I say- it’s still high, Sir."

Election Results 2024 Live: BJP wins Banglore South

Election Results 2024 Live: BJP Yuva Morcha national president & sitting MP, Tejasvi Surya wins from Karnataka's Banglore South Lok Sabha constituency.

Election Results 2024 Live: BJP defeated in Ayodhya constituency

Election Results 2024 Live: BJP conceded defeat on Tuesday in the constituency where the inauguration of Ram Mandir just months ago by PM Modi was seen as an event that would cement his legacy and the party's poll victory.

The BJP looks set for heavy losses in the bellwether northern state of Uttar Pradesh, as the opposition Samajwadi Party and Congress race ahead in more than half of its 80 seats, including in Ayodhya city in the Faizabad constituency where Modi inaugurated the temple in January.

Election Results 2024 Live: Priyanka Gandhi praises people of UP, says ‘most proud’

Election Results 2024 Live: Congress leader Priyanka Gandhi Vadra said, "I am very happy... I want to tell the people of UP that they have shown a lot of discretion. I am most proud of UP..."

Election Results 2024 Live: CM Mamata Banerjee seeks PM Modi's ‘immediate resignation’

Election Results 2024 Live: Seeking PM Modi's 'immediate resignation", West engal CM Mamata Banerjee said that she would try to ensure that Modi “is out of power and INDIA bloc is in" following the results. 

The Trinamool Congress chief claimed that the people of India had ‘broken’ PM Modi's morale after he ‘broke’ many politicial parties.

Election Results 2024 Live: BJP leading in all Delhi-NCT seats 

Election Results 2024 Live: BJP is leading in all seven seats of Delhi-NCT. These seven seats include – Chandni Chowk, East Delhi, New Delhi, North-East Delhi, South Delhi, North-West Delhi, West Delhi. 

In 2019, BJP had won all seven seats in Delhi Lok Sabha constituencies. 

Election Results 2024 Live: Arun Govil wins from Meerut 

Election Results 2024 Live: Ramayan fame actor Arun Govil wins from Meerut Lok Sabha constituency seat in Uttar Pradesh. He contested against Samajwadi Party candidate Sunita Verma.

Govil got 546469 votes in Meerut.

Election Results 2024 Live: Rahul Gandhi is all praise for UP voter base

Election Results 2024 Live: Congress leader Rahul Gandhi says, "UP ki janta ne kamaal karke dikha diya...The people of UP understood the politics of the country and the danger to the Constitution, and they safeguarded the Constitution. I thank them for supporting Congress party and INDIA alliance..."

Election Results 2024 Live: Smriti Irani breaks silence after loss, says ‘I will continue to be…’

Election Results 2024 Live: Union Minister and BJP's candidate from Uttar Pradesh's Amethi Lok Sabha seat, Smriti Irani says, "...I express my gratitude to all the BJP party workers and supporters, those who have worked in the service of the constituency and the party with utmost dedication and fidelity...Today, I am grateful to PM Narendra Modi and CM Yogi Adityanath that their governments have completed the pending works of 30 years in just 5 years. I congratulate those who win. I will continue to be in the service of the people of Amethi..."

Election Results 2024 Live: Hassan MP Prajwal Revanna loses by a margin of 42,649 votes.

Election Results 2024 Live: Hassan MP Prajwal Revanna, accused of sexually assaulting several women, and facing a SIT probe for the same, lost by a margin of 42,649 votes.

Election Results 2024 Live: Rahul will not stay for Rae Bareli and Wayanad seats

Election Results 2024 Live: Congress leader Rahul Gandhi said that he won Wayanad in Kerala and Rae Bareli in Uttar Pradesh. He mentioned that he would not be able to stay MP for both Lok Sabha Constituency. 'Cant stay on both seats, haven't decided', said Rahul Gandhi. 

Election Results 2024 Live:  Latest trends as per ECI

Bharatiya Janata Party -  58 (won) 183 (leading) 

Indian National Congress - 22 (won) 76 (leading)

Election Results 2024 Live: INDIA bloc to meet tomorrow 

Election Results 2024 Live: INDIA bloc to meet tomorrow to discuss government formation, confirms Rahul Gandhi.

Election Results 2024 Live: Rahul Gandhi flags Adani stocks

Election Results 2024 Live: Rahul Gandhi says Adani stocks tumled onTuesday because the nation does not want PM Modi. 

Election Results 2024 Live: Rahul Gandhi thanks all ‘Babbar Sher’

Election Results 2024 Live: Rahul Gandhi thanks INDIA bloc allies. The Congress leader said that wherever the INDIA loc fought, it won. The Congress leader also thanked the ‘Babbar Sher’ of the party. 

Election Results 2024 Live: Fought against ED, CBI, says Rahul Gandhi

Election Results 2024 Live: Rahul Gandhi said that the Congress fought to save the Constitution, and that the Granbd OLd Party fought against tyhe ED and CBI which were ‘captured and threatened’ by  PM Narendra Modi and BJP leaeder Amit Shah.

Election Results 2024 Live: SP candidate Dimple Yadav wins in Mainpuri

Election Results 2024 Live: Akhilesh Yadav's wife, Dimple Yadav wins Uttar Pradesh's Mainpuri

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