2000/04/01 by Daniel Shiu · 2 citations
Mathematics · #History and Theory of Mathematics #Conjecture #Combinatorics #Mathematics #Modulo #Mod #Prime (order theory) #Discrete mathematics
paper · doi:10.1112/s0024610799007863
openalex publication_date 2000/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
In 1920 Chowla made the following conjecture. Let pn denote the nth prime; if q ⩾ 3, (q, a) = 1 then there are infinitely many pairs of consecutive primes pn and pn+1 such that pn ≡ pn+1 ≡ a mod q By considering the sum ∑ p χ ( p ) 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 pn+1≡pn+2≡…≡pn+k≡a mod q For each modulus q the method used applies best to the following two sets of residue classes: A+:=a:∀ p|q≡1 mod p A−:=a:∀ p|q,a≡−1 mod p Larger values of k in terms of pn+1 can be found for residue classes belonging to these sets.