Residue class patterns of consecutive primes

Dickson’s conjecture and the Hardy–Littlewood prime tuple conjecture predict that every pattern of reduced residue classes modulo q is attained by infinitely many strings of m consecutive primes. At present, however, even proving that a single non-constant residue class pattern of length m occurs infinitely often is beyond the reach of existing methods. Combining Dirichlet’s theorem on primes in arithmetic progressions with a theorem of Shiu (2000) shows that, for any m,q∈N with q≥3, at least mφ(q) residue class patterns of length m are attained by infinitely many consecutive primes. We prove that if q is squarefree, every prescribed sequence of at least 60mlogm reduced residue classes mod q contains, in order, an m-term block pattern that occurs infinitely often among consecutive primes, with each constant block of length at most ⌈logm⌉. A recursive combinatorial argument then shows that if q is squarefree and q≫(logm)2, then at least ≫m(logm)10φ(q)2 residue class patterns of length m occur infinitely often among consecutive primes. Moreover, we also show that if q is squarefree and q≫(logm)2, then at least ≫e−O(mlog2m/logm)φ(q)m/⌈logm⌉ residue class patterns of length m occur infinitely often among consecutive primes. The proof consists of a modification of the Maynard–Tao sieve found in Banks, Freiberg, and Maynard (2016), by considering the r-th moment instead of the 2nd moment for an integer r depending on m, which is then combined with an Erdős–Rankin type construction.