Consecutive integers for which n² + 1 is composite

Betty Kvarda

Vol. 97 (1981), No. 1, 93–96

DOI: 10.2140/pjm.1981.97.93

Abstract

Let 𝒫 = {p_k}_k=0^∞ where p₀ = 2 and p_k, k > 0, is the k-th prime in the sequence of positive integers congruent to 1 modulo 4. Thus 𝒫 contains the prime divisors of all the integers n² + 1. For each t = 0,1,⋯ let P(t) = Π_k=0^tp_k. It will be shown that for each sufficiently large integer t there exists a sequence 𝒞_t of consecutive integers n such that (i) (n² + 1,P(t)) > 1 for all n in 𝒞_t, (ii) card 𝒞_t ≧ [(1 −𝜀)λp_t], 0 < 𝜀 < 1, for a certain positive constant λ, and (iii) p_t < n < P(t) for all n in 𝒞_t.

Mathematical Subject Classification

Primary: 10A40, 10A40

Milestones

Received: 25 March 1980

Revised: 27 March 1981

Published: 1 November 1981