Computers As a Novel Mathematical Reality: IV. The Goldbach Problem

In this part I pursue the discussion of the role of computers in additive number theroy. I sketch the definitive solution of the ternary = odd Goldbach conjecture, not in one of its 20th century asymptotric reformulations, but in its original 17th century form. Namely, that every odd number n > 5 is a sum of three positive rational primes. This problem was only solved in 2013–2014 by Harald Helfgott, and there is no chance that it could have been obtained without the use of computers. Apart from that, I discuss the status of the binary = even Goldbach conjecture and partial results leading towards its solution, as well as some further related problems.