*** Fundamental Theorem of Arithmetic ***
This is the most basic of all theorems. In its simplest form, the theorem says that every positive integer can be "split" into prime factors in only one way. In other words, the prime factors of any number are unique. For example, 1386 = 2 x 3 x 3 x 7 x 11 5A = 2 x 2 x 3 x 5 The theorem says that these are the only such representation using prime factors.*** Fermat's Last Theorem ***
This theorem states that a^n + b^n = c^n is solvable in integers a,b,c only when n is 1 or 2. The case with n=2 are commonly known as Pythagorean Triples such as 3^2 + 4^2 = 5^2. Until recently, this theorem was really a conjecture. It had been proven for many cases of n, but not generally for all n to infinity. In 1994, Andrew Wiles proved it.*** Fermat's Lesser Theorem ***
This theorem says that (b^(p-1) - 1) has p as a factor when b and p are relatively prime, and p is prime. For example, 5^6 - 1 = 2 x 2 x 2 x 7 x 279 and has 7 as a factor However there are cases when p is composite such as b=2 and p = 341 = 11 x 31 In other words, (2^340 - 1) has 341 as a factor. Fermat's Lesser Theorem was generalized by Euler in terms of a totient function to replace the power (p-1).*** Wilson's Theorem ***
This theorem states that (p-1)!+1 has p as a factor when p is prime. For example, 6!+1 = 721 = 7 x A3 which has 7 as a factor. Gauss generalized Wilson's Theorem.*** Goldbach's Conjecture ***
This is still unresolved, and states that every even integer greater than or equal to 6, can be represented as the sum of two odd primes. So, for example, 6 = 3 + 3, 8 = 3 + 5, 28 = 5 + 23, 96 = 7 + 89 Also odd numbers >= 9, are the sum of three odd primes. However this is easily reducible to the case for even numbers. A related conjecture to this theorem is to say that every integer >= 4, lies midway between two odd primes: 4 lies midway between 3 and 5, 5 lies midway between 3 and 7, 6 lies midway between 5 and 7, 7 lies midway between 3 and 11, 8 lies midway between 5 and 11 as well as 3 and 13, 9 lies midway between 7 and 11 as well as 5 and 13, and so on... As the integer increases, the number of such pairs generally increases, however, it has still not been proven that there are such cases for all integers.
Enjoy.