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Let p_n denote the sequence of all primes and let d_n=p_n-p_{n-1} denote thesequence of all gaps between consecutive primes. In 1948 Erd\H{o}s and Tur\'anshowed that d_{n+1}-d_n changes sign infinitely often and together with P\'olyaasked for a necessary and sufficient condition that a fixed linear combinationof r>1 consecutive prime gaps d_{n+i} i=(1,2,...r) should change signinfinitely often as n runs through the sequence of all natural numbers. Theyconjectured that the condition is that the non zero values among thecoefficients of the linear combination cannot all have the same sign but theycould prove only much weaker relations. In the present work this conjecture isproved by the aid of the Maynard-Tao method and other important ideas of therecent work of W.D. Banks, T. Freiberg and J.Maynard (see arXiv:1404.59094v2).

On a conjecture of Erd\H{O}s, P\'olya and Tur\'an on consecutive gaps between primes