Square-free values of n²+1

Abstract

Schinzel in celebration of his seventy-fifth birthday 1. Introduction.Let N (x) denote the number of positive integers n ≤ x for which n 2 + 1 is square-free.It was shown in 1931 by Estermann [4] that N (x) = c 0 x + O(x 2/3 log x) for x ≥ 2, where c 0 = 1 2 p≡1 mod 4(1 -2p -2 ).Estermann's argument is very simple, but despite the passage of 80 years the exponent 2/3 appearing above has never been improved.The aim of the present paper is to establish the following result.Theorem.We havefor any fixed ε > 0.It is easy to construct intervals (x, x + c log x] with a small positive constant c, such that n 2 + 1 has a non-trivial square factor for every n in the interval.This shows that the error term in our theorem is Ω(log x).However we know of no better result of this type, and it is unclear what one should conjecture.With the much simpler problem of the number of square-free integers n ≤ x one has an easy error term O(x 1/2 ), but any reduction in the exponent 1/2 would appear to require a quasi Riemann Hypothesis.Thus it seems unlikely that we could reduce the exponent in our theorem below 1/2 without a radically new idea.

Locations

• Acta Arithmetica - View - PDF