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
formula here
By considering the sum
formula here
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
formula here
For each modulus q the method used applies best to the following two sets of residue classes:
formula here
Larger values of k in terms of pn+1 can be found for
residue classes belonging to these sets.