(See the sibling comment with a "10." in it for context.) 9. The weak Goldbach conjecture. Back in 1742, Christian Goldbach (like Fermat, a mathematician-and-l

(See the sibling comment with a "10." in it for context.) 9. The weak Goldbach conjecture. Back in 1742, Christian Goldbach (like Fermat, a mathematician-and-lawyer who made a famous long-unsolved conjecture in number theory) wrote in a letter to Euler that he thought every even number bigger than 2 could be written as a sum of two prime numbers. (E.g., 4=2+2; 6=3+3; 8=3+5; 10=3+7; 12=5+7.) Everyone believes this is true; there are good heuristic arguments why it "should" be true; but proving it is really hard. Like the abc conjecture, this is another one about the interplay between the additive and multiplicative structure of the integers. One method that's been used to attack the Goldbach conjecture is the "Hardy-Littlewood circle method", which is useful for a whole lot of other kinda-similar problems in number theory; it's kinda-sorta an application of Fourier transforms to number theory. It was used back in the 1930s to prove the following statement, which is weaker in two ways: All large enough odd numbers are the sum of three prime numbers. (Note that if GC is true then so is this; if even=prime+prime then even+3=prime+prime+3.) What Helfgott has now done is to show that "large enough" means "bigger than 5" (which is the best that could possibly be done, since 5 is too small to be the sum of three primes). He did it by refining the "circle method". Getting from here to the full Goldbach conjecture is still going to require major new ideas; there are somewhat-well-understood obstacles to using this method to prove GC itself.