Prime Numbers

Watch

• The Oldest Unsolved Problem in Math ↗Veritasium
• We're 99.9% sure this pattern is true, but no one can prove it ↗Veritasium
• What are Prime Numbers? w/Alan Davies | Why Maths Doesn't Add Up | BBC Studios ↗BBC Studios
• Are Prime Numbers Made Up? | Infinite Series | PBS Digital Studios ↗PBS Infinite Series
• Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations ↗3Blue1Brown
• The Most Wanted Prime Number - Numberphile ↗Numberphile

Read

• The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics ↗A captivating narrative introducing the allure and enduring enigma of prime numbers, accessible to the mathematically curious.Marcus du Sautoy
• Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics ↗Explores the profound implications of the Riemann Hypothesis, offering a compelling journey into the heart of modern number theory.John Derbyshire
• An Introduction to the Theory of Numbers ↗The definitive classical treatise, essential for any serious student seeking a rigorous foundation in number theory.G.H. Hardy and E.M. Wright
• On the Number of Prime Numbers Less Than a Given Magnitude (Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse)The seminal work that introduced the Riemann Hypothesis, forever shaping the quest to understand prime distribution.Bernhard Riemann
• What is Mathematics? An Elementary Approach to Ideas and Methods ↗Offers an accessible yet profound exploration of mathematical concepts, with an illuminating section on prime numbers and their properties.Richard Courant and Herbert Robbins

Voices to follow

• Andrew Wiles ↗Celebrated for his proof of Fermat's Last Theorem, a monumental achievement that illuminated deep connections within number theory, including the study of primes.Royal Society Research Professor, University of Oxford; Professor Emeritus, Princeton University
• Terence Tao ↗Known as the 'Mozart of Math,' his groundbreaking work in analytic number theory, including the Green-Tao theorem on prime number progressions, continues to reshape the field.Professor of Mathematics, University of California, Los Angeles; Fields Medalist
• Marcus du Sautoy ↗An engaging populariser, he brings the mysteries of prime numbers and the Riemann Hypothesis to a wide audience through his books and broadcasts, making complex ideas accessible.Professor of Mathematics, University of Oxford; Simonyi Professor for the Public Understanding of Science
• Vicky Neale ↗An eloquent communicator, she demystifies the beauty and challenges of number theory, including the enigmatic world of primes, through her lectures and writing.Whitehead Lecturer, Mathematical Institute, University of Oxford

Glossary

• Co-prime (or Relatively Prime)Two numbers are co-prime (or relatively prime) if their only common factor is 1. This means they don't share any other prime factors. For example, 8 and 9 are co-prime because the factors of 8 are 1, 2, 4, 8 and the factors of 9 are 1, 3, 9; the only factor they share is 1.
• Composite NumberA composite number is a whole number greater than 1 that has more than two factors (numbers that divide it evenly). Unlike prime numbers, it can be divided by numbers other than 1 and itself. For example, 6 is a composite number because it can be divided evenly by 1, 2, 3, and 6.
• Factor (or Divisor)A factor (also called a divisor) is a whole number that divides another number exactly, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, because each of these numbers divides 12 perfectly.
• Fundamental Theorem of ArithmeticThis important theorem states that every whole number greater than 1 is either a prime number itself or can be written as a unique product of prime numbers (ignoring the order of the factors). It means there's only one way to break a number down into its prime building blocks. For example, the number 30 can only be written as 2 x 3 x 5 (or 3 x 2 x 5, etc.), and never as any other set of prime numbers.
• Greatest Common Divisor (GCD) / Highest Common Factor (HCF)The Greatest Common Divisor (GCD), also known as the Highest Common Factor (HCF), is the largest number that divides two or more numbers exactly without leaving a remainder. It's the biggest factor they share. For example, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 evenly.
• Least Common Multiple (LCM)The Least Common Multiple (LCM) is the smallest positive number that is a multiple of two or more numbers. It's the first number they "meet" at when you list their multiples. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that is a multiple of both 4 (4, 8, 12) and 6 (6, 12).
• MultipleA multiple of a number is the result you get when you multiply that number by another whole number. It's like counting by that number. For example, the multiples of 3 are 3, 6, 9, 12, and so on, because 3x1=3, 3x2=6, 3x3=9, etc.
• Prime FactorizationPrime factorization is the process of breaking down a composite number into its prime number factors. It's like finding the "building blocks" of a number using only prime numbers. For example, the prime factorization of 12 is 2 x 2 x 3, because 2 and 3 are prime numbers that multiply to 12.
• Prime NumberA prime number is a whole number greater than 1 that can only be divided evenly by 1 and itself. It has no other factors (numbers that divide it without a remainder). For example, 7 is a prime number because you can only divide it evenly by 1 and 7.
• Sieve of EratosthenesThe Sieve of Eratosthenes is an ancient method for finding all prime numbers up to any given limit. You start by listing numbers and then systematically cross out multiples of each prime number you find until only the prime numbers are left. For example, to find primes up to 10, you'd list 2-10, circle 2 and cross out its multiples (4, 6, 8, 10), then circle 3 and cross out its multiples (6, 9), leaving 2, 3, 5, 7 as primes.

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Where this connects to other fields, and why it's worth knowing.

• Money, Banking & Credit Economics

  Banks exist partly so two strangers can trust a payment. Prime-number math does the same job without any middleman: it lets people who've never met send money safely, building trust out of pure arithmetic instead of a bank vouching for them.

• Epistemology Philosophy

  There's a clever trick called a zero-knowledge proof: you can prove you know a password without ever revealing the password. That turns a deep question, 'how do I know you know?', into pure math. You pass along the certainty without passing along the secret itself.

• Writing Systems Literature

  Cracking a lost ancient script and cracking a secret code are the same puzzle: hunt for patterns in symbols. That's why Champollion decoding Egyptian hieroglyphs and Turing breaking Nazi codes were really doing one craft, two centuries apart.

• Elections & Voting Political Science

  Voting seems stuck with a trade-off: keep your ballot secret, or let people verify the count, but not both. Clever math from internet security can do both at once, proving your vote was counted without revealing who you picked. It dissolves a problem democracy long thought was unavoidable.