On a Conjecture of Erdős, Pólya and Turán on Consecutive Gaps Between Primes

Erdős, Pólya and Turán conjectured 70 years ago that a linear combination of consecutive differences of primes takes infinitely often both positive and negative values if and only if the (fixed) coefficients of the linear combination do not have all the same sign. In this work we prove this conjecture in a somewhat more general form. Our proof is based on a method of Banks, Freiberg and Maynard which is again based on the method of Maynard, Tao and the Polymath 8 project which showed the existence of infinitely many prime gaps not exceeding 246.