Prime numbers theory
The fundamental theorem of arithmetic states that any positive integer can be represented in exactly one way as a product of primes. Here is a short prime numbers list: 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, 101. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way. Euclid also showed that if the number 2^n - 1 is prime then the number 2^n-1(2n - 1) is a perfect number.
Eratosthenes (275-194 B.C., Greece) devised a 'sieve' to discover prime numbers. A sieve is like a strainer that you use to drain spaghetti when it is done cooking. Eratosthenes's sieve drains out composite numbers and leaves prime numbers behind.