Terence Tao has submitted a paper to arxiv: [1201.6656] Every odd number greater than 1 is the sum of at most five primes Its abstract: One can turn the Goldbach conjecture and similar problems into statements about certain integrals, but those integrals are VERY hard to do, and it has only been possible to find analytic bounds for those integrals for very large numbers. There are, in fact, 2 Goldbach conjectures Even: every even number x > 0 is the sum of at most 2 primes Odd: every odd number x > 1 is the sum of at most 3 primes The even one implies the odd one, and they both imply the 5-prime one, but these implications do not work in reverse. TT quotes: Chen and Wang for odd: x >= exp(exp(11.503)) ~ exp(99000) ~ 3.33*1043000 Liu and Wang for odd: x >= exp(3100) ~ 2.1*101346 Richstein for even: x <= 4*1014 ~ exp(33) Ramaré and Saouter for odd: x <= exp(28) ~ 1.14*1022 ~ exp(51) (their paper) So there's a big gap between the analytic results and the numerical ones. TT's paper is about closing that gap for the 5-prime case, and I will concede that I find it difficult to follow. But if his reasoning is sound, then that suggests that we may not be far off from proving Goldbach's conjecture by a similar method, or else making it much easier to find a counterexample. Could it be possible to find bounds from above for similar unsolved mathematical problems? The Riemann hypothesis. It's about the zeros of the Riemann zeta function, the values that make it zero. Its trivial zeros are the negative even integers, and the rest are its nontrivial ones. The hypothesis states that all of them have real part 1/2. Mersenne primes. Mersenne numbers 2prime-1 include primes as far as they have been searched: 243,112,609 – 1 is the largest currently known one. It is not known whether there is a finite or infinite number of them. Fermat primes. Fermat numbers 22^n+1 include only 5 known primes: n = 0 to 4: 3, 5, 17, 257, 65537. For n = 5 to 32, the Fermat numbers are known to be composite, and it is not known whether or not there are any other Fermat primes.