Goldbach’s Conjecture was first proposed in a weak form by Christian Goldbach in the 18th century and then in a stronger form by Leonhard Euler. It says that every odd number can be broken up into a sum of, at most, 3 prime numbers.

35 = 19 + 13 + 3

21 = 3 + 7 + 11

It has not been proven so far, but the young Terence Tao seems to be making progress.

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Can it be proven that every even is the sum of two or less primes?

The strong Goldbach Conjecture, proposed by Euler, says that every even number larger than 2 is the sum of two primes.

Neither one has been proven, but a computer did verify the validity of both for numbers up to 19 digits long.