Goldbach’s conjectures: A historical perspective

Abstract

In 1742, Goldbach and Euler in conversation and in an exchange of letters discussed the representation of numbers as sums of at most three primes. Although the question as to whether every even number is the sum of one or two primes (the binary Goldbach conjecture) is still unresolved, this and associated questions have attracted many mathematicians over the years, and have lead to a range of powerful techniques with many applications. This article is a commentary on the historical developments, the underlying key ideas and their widespread influence on a variety of central questions.