Computers as Novel Mathematical Reality. IV. Goldbach Problem

Abstract

In this part I pursue the discussion of the role of computers in additive number theroy. Here I sketch the definitive solution of the ternary = odd Goldbach problem, not in one of its XX century asymptotric reformulations, but in its original XVII century form. Namely, that every odd number n > 5 is a sum n = p1 + p2 + p3 of three positive rational primes. A solution of this problem was only completed by Harald Helfgott in 2013–2014 and there is no chance that it could be obtained without the use of computers. Apart from that, I discuss the status of the binary = even Goldbach problem, partial results towards its solution, as well as some further related proiblems.