Goldbach's conjecture: every even integer greater than 2 can be expressed as the sum of two primes.
Since every uneven number is x+1, where x is an even number, and every such number can be expressed as the sum of 3 even integers, namely x + y + 2, where x and y are even numbers, or 2n + 2m + 2, which can be expressed as a summation of twos (2+2+...) or simply 2p, which by definition is divisible by two and therefore is an even number, we can conclude that since every prime number is uneven, except 2, this proves that every even number, except 4, can be obtained by 2 prime numbers.
Every summation of two prime numbers can be expressed as 2n + 2m + 2, since it is always divisible by 2. Thus any prime number is the summation of an even number n and 1. This can be easily seen in the simple fact that 2 is and can be the only even prime number. Think of an even number as a summation of twos (2+2+...). It is always divisible by 2, 1 and itself. Thus we have proven that any prime number cannot be an even number, except 2.
By exhaustion, we can prove the conjecture. We only need to prove it for 4, since we've already proven it for all other even integers.
We simply remove 2m from the equation, which gives 2n + 2, which comes from 2n + 1 +1, where n must be 1. 3 and 1 are primes as required, thus we have proven the conjecture.