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.
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.