Strings of congruent primes in short intervals
arXiv:1005.4703 · doi:10.1112/jlms/jdr006
Abstract
Fix ε> 0, and let p_1 = 2, p_2 = 3,... be the sequence of all primes. We prove that if (q,a) = 1 then there are infinitely many pairs p_r, p_{r+1} such that p_r \equiv p_{r+1} \equiv a \mod q and p_{r+1} - p_r < ε\log p_r. The proof combines the ideas of Shiu and Goldston-Pintz-Yildirim.
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