Vol. 97, No. 1, 1981

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Consecutive integers for which n2 + 1 is composite

Betty Kvarda

Vol. 97 (1981), No. 1, 93–96
Abstract

Let 𝒫 = {pk}k=0 where p0 = 2 and pk, 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) = Πk=0tpk. It will be shown that for each sufficiently large integer t there exists a sequence 𝒞t of consecutive integers n such that (i) (n2 + 1,P(t)) > 1 for all n in 𝒞t, (ii) card 𝒞t [(1 𝜀)λpt], 0 < 𝜀 < 1, for a certain positive constant λ, and (iii) pt < n < P(t) for all n in 𝒞t.

Mathematical Subject Classification
Primary: 10A40, 10A40
Milestones
Received: 25 March 1980
Revised: 27 March 1981
Published: 1 November 1981
Authors
Betty Kvarda