Search

Let S be a nonempty finite subset of ℕ and : S → { − 1 , 1 } be an arbitrary map (choice of signs for S). We will say that  S  has residue pattern   modulo  p  if  s p = (s)   ∀ s ∈ S , where ⋅ p is the Legendre symbol mod  p. For a given nonempty finite subset S of ℕ with a choice of signs and a real number 1 ≤ c < 1 2 1 1 , we obtain an asymptotic formula for the number of primes p of the form p = ⌊ n c ⌋ such that S has residue pattern modulo p which also satisfies p ≡ f  (mod  d) where f , d are integers with 1 ≤ f ≤ d and (f , d) = 1. For an irrational α > 1 and a real β ≥ 0 , we also obtain an asymptotic formula for the same but primes p of the form p = ⌊ α n + β ⌋ under certain assumption on α. [ABSTRACT FROM AUTHOR]