Landau’s problems on primes

Abstract

At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following: (1) Are there infinitely many primes of the form n2 + 1? (2) The (Binary) Goldbach Conjecture, that every even number exceeding 2 can be written as the sum of two primes. (3) The Twin Prime Conjecture. (4) Does there exist always at least one prime between neighbouring squares? All these problems are still open. In the present work a survey will be given about partial results in Problems (2)-(4), with special emphasis on the recent results of D. Goldston, C. Yildirim and the author on small gaps between primes.