The conjecture states:
Every even integer greater than 2 is the sum of 2 primes.
How do we prove it? Nobody really has that figured out yet.
It’s simple to try every single number individually to see if this conjecture holds true, but that does not give proof that it is true for every single number ever in existence. Many mathematicians have spent time pushing the bounds of this conjecture to higher and higher numbers, and have even tried to prove a weaker version of this conjecture.
The weaker version states that all odd numbers greater than nine are the sum of 3 odd primes. The strong version states that all even integers greater than 4 can be broken down into the sum of two primes. The Goldbach Partition is then written as: 2n = p + q, where n is a positive integer and p and q are both prime numbers.
Terence Tao proved that every integer greater than 1 is the sum of no more than 5 prime numbers, which is a step in the right direction to proving this conjecture true. Vinogradov has proven that all sufficiently large odd numbers work out to be a sum of 3 primes, and Estermann has proven that almost all even numbers are the sum of two primes.
So far, no one has completely proven Goldbach’s true conjecture.