Pintz János: On a conjecture of Descartes

Descartes conjectured a century before Goldbach a similar, but different conjecture. According to this, every even number can be written as the sum of at most three pimes. It is easy to see that this is equivalent with the conjecture that for every even N at least one of N and N+2 is the sum of two primes. At present the conjecture seems to be hopeless. (The best result in this direction is that at least one of N, N+2, N+4... N+M can be written as the sum of two primes where M=N^b for a b approximately 1/20.)  This implies that the size D(X) of the exceptional set for Descartes conjecture (for even numbers below a large number X) is at most of the size E(X) of the exceptional set for the Goldbach conjecture. Earlier methods were unable to estimate D(X) better than E(X). We sketch the proof that for any c>3/5 we have D(X)= O(X^c) which is stronger than the best results for E(X).