Goldbach's Problem

This is an exposition of several important results on the Goldbach conjecture. That conjecture, proposed by Christian Goldbach in a 1742 letter to Euler, is now viewed as being in two parts. The binary Goldbach conjecture asserts that every even number \(> 2\) is the sum of two primes. The ternary Goldbach conjecture asserts that every odd number \(> 5\) is the sum of three primes. These problems are sufficiently different that most work has been on one or the other of them. The present book deals only with the ternary Goldbach problem, except for an early result of Schnirelmann that applies to both.

The proofs in this book are very clearly written, but in general are not standalone proofs because they depend on deep results from prime number theory that are quoted without proof. The first chapter is an exposition of I. M. Vinogradov’s 1937 proof that all sufficiently large odd numbers satisfy the ternary Goldbach conjecture. The proof uses Vinogradov’s theory of trigonometric sums and his version of the Hardy-Littlewood circle method. The proof assumes the Siegel–Walfisz theorem.

The second chapter gives a proof of Maier and Rassias that, assuming the Generalized Riemann Hypothesis (GRH), every sufficiently large even number is the sum of a prime and of two “isolated” primes (isolated here means, in a precise sense, that no other primes are near them). The proof builds on Vinogradov’s proof from the first chapter, and draws (without proof) on a number of sieve results and on more precise weighted prime sums depending on the GRH.

The last chapter goes back in time to 1930 and gives Schnirelmann’s proof that every sufficiently large integer (even or odd) is the sum of a bounded number of primes. Schnirelmann determined the bound to be \(< 800,000\) primes, which is much better than the previous bound of infinity but still not very close to the conjectured result. Schnirelmann’s result is important because it introduced the concept of Schnirelmann’s density, which has been useful in many additive problems. The proof is also interesting because, unlike the results mentioned above, it uses only elementary number theory and combinatorial reasoning.

Harald Helfgott in 2013 announced an unconditional proof of the ternary Goldbach conjecture and posted it on the arXiv, but it has not been published in a journal yet. The present book includes a sketch of the proof (contributed by Olivier Ramaré). The proof also follows Vinogradov’s lead, and shows the result is true for all numbers greater than \(10^27\) and uses a computer search to verify the smaller ones.

Another good book along these lines is Nathanson’s Additive Number Theory: The Classical Bases. This is older and not as up to date, but it does give a very detailed exposition, covering both the binary and ternary Goldbach conjectures, and also covers the Waring problem. A valuable historical book is Yuan Wang’s collection The Goldbach Conjecture, which reprints most of the important research papers in the subject, either in the original English or in English translation.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.