In 1920 Chowla made the following conjecture. Let pn denote the nth prime; if q [ges ] 3, (q, a) = 1 then there are infinitely many pairs of consecutive primes pn and pn+1 such that

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By considering the sum

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where χ is the non-principal character modulo 4 or 6, it is possible to prove the conjecture for q = 4 and q = 6 (a = ±1). In this paper we prove Chowla's conjecture for all q and a with (q, a) = 1. Moreover, we shall show that for any k there exist ‘strings’ of congruent primes such that

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For each modulus q the method used applies best to the following two sets of residue classes:

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Larger values of k in terms of pn+1 can be found for residue classes belonging to these sets.