The values of a quadratic form at square-free points

Values of isotropic quadratic forms at S-integral points

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Values of Quadratic Forms at Integral Points : an Elementary Approach

On Symmetric Square Values of Quadratic Polynomials

We prove that there does not exist a non-square quadratic polynomial with integer coefficients and an axis of symmetry which takes square values for N consecutive integers for N = 7 or N ≥ 9. At the opposite, if N ≤ 6 or N = 8 there are infinitely many.

Square free values of the order function

Given a ∈ Z \ {±1, 0}, we consider the problem of enumerating the integers m coprime to a such that the order of a modulo m is square free. This question is raised in analogy to a result recently proved jointly with F. Saidak and I. Shparlinski where square free values of the Carmichael function are studied. The technique is the one of Hooley that uses the Chebotarev Density Theorem to enumerat...

Square-free values of the Carmichael function

We obtain an asymptotic formula for the number of square-free values among p 1; for primes ppx; and we apply it to derive the following asymptotic formula for LðxÞ; the number of square-free values of the Carmichael function lðnÞ for 1pnpx; LðxÞ 1⁄4 ðkþ oð1ÞÞ x ln a x ; where a 1⁄4 0:37395y is the Artin constant, and k 1⁄4 0:80328y is another absolute constant. r 2003 Elsevier Inc. All rights r...