3.1 Prime and Composite Numbers — Illegal Numbers Secure the Internet | Contemporary Mathematics

• Understand integers, natural numbers, and the concept of divisibility without remainder.
• Learn the mathematical mechanisms behind divisibility rules for 3, 9, and 4.
• See how divisibility applies to practical scenarios like distributing items equally.
• Understand the definitions of prime and composite numbers and the special case of one.
• Learn the efficient method of checking for prime numbers by testing divisors up to the square root.
• Discover how massive prime numbers form the foundation of digital security through trapdoor functions.
• Explore how specific prime numbers can become illegal due to their connection to copyrighted code.
• Learn about Sophie Germain's contributions to number theory and the definition of Sophie Germain primes.
• Discover Ramanujan's unique ability to perceive the intricate structures and 'personalities' of numbers.
• Understand that every composite number has a unique prime factorization, like a chemical formula.
• Learn how to find the largest common divisor between numbers using prime factorization for practical organization.
• Understand how the LCM helps synchronize independent cycles and predict when events will overlap.
• Explore how cicadas use prime number life cycles as an evolutionary strategy to avoid predators.
• Up next in “Contemporary Math — Full Course (OpenStax)”3.2 The Integers — Why Negative Numbers Are a Superpower | Contemporary Mathematics

Discover why certain prime numbers are so valuable they've been declared illegal — and how the same math that stumped ancient scholars now protects your online banking. This lesson covers Section 3.1 of OpenStax Contemporary Mathematics, exploring divisibility, prime and composite numbers, and their surprisingly high-stakes real-world applications. From shortcuts for testing divisibility to the 300-trillion-year encryption standard, the concepts here reveal the hidden architecture behind everyday digital life. In this video: • Divisibility rules for 2, 3, 4, 5, 6, 9, and 10 — including why adding a number's digits reveals divisibility by 3 and 9 • Prime vs. composite numbers explained with examples like 19 and 26, plus the special status of the number 1 • The square root shortcut: why you only need to test prime divisors up to the square root of a number (demonstrated with 2,117) • RSA-style encryption as a 'trapdoor function' — multiplying two large primes takes milliseconds, but reversing it would take a computer 300 trillion years • Illegal primes: how a 1,400-digit prime number was declared contraband under the Digital Millennium Copyright Act • Sophie Germain and Ramanujan — their contributions to prime number theory, including Sophie Germain primes (e.g., 23 → 47) and the taxicab number 1,729 • GCD and LCM applied to tiling a 570×450 cm room, synchronizing volunteer schedules, and cicadas evolving prime-number life cycles as evolutionary cryptography #OpenStax #PrimeNumbers #ContemporaryMathematics #Encryption #NumberTheory OpenStax Content adapted from "Contemporary Math", by OpenStax Licensed under CC BY 4.0. Content based on Web Version: Apr 23, 2026. Read the textbook online https://openstax.org/details/books/contemporary-mathematics Music first girl talking to me. by ikkun (ex. Barradeen) | https://soundcloud.com/ikkunwastaken Royalty Free Music by https://www.free-stock-music.com Creative Commons / Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) https://creativecommons.org/licenses/by-sa/3.0/deed.en_US