The Beauty of Integers

Number Theory · Axiom Academy

Discover why the simple counting numbers hold mysteries that have captivated mathematicians for millennia. Move the slider to explore integers. Watch as a mysterious pattern emerges among certain numbers. Perfect Numbers: Ancient Perfection Click on different numbers to see their divisors. Some numbers have a special property known since ancient Greece. Simple Questions, Deep Mysteries Click on each conjecture to reveal an unsolved mystery about integers. These simple questions have no known answers. Conjecture: Every even integer greater than 2 can be written as the sum of two primes. Examples: 4 = 2+2, 6 = 3+3, 8 = 3+5, 10 = 3+7 = 5+5 Conjecture: There are infinitely many pairs of primes that differ by 2. Examples: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43)... The largest known twin primes have over 388,000 digits each! But no one has proven infinitely many exist. Rule: Start with any positive integer. If even, divide by 2. If odd, multiply by 3 and add 1. Repeat. Conjecture: You always eventually reach 1. Example (7): 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 Verified for all numbers up to 2 68 , but no proof exists! Carl Friedrich Gauss called number theory "the queen of mathematics." Discover why.

This is the written version of the interactive lesson above. See the full Number Theory course.