Informatics Publishing Limited

Florian Luca ¹, Igor E. Shparlinski ², R. Thangadurai ³

Affiliations
1 Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, C.P. 58089, Morelia, Michoacan, MX
2 Department of Computing, Macquarie University, Sydney, NSW 2109, AU
3 Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, IN

Abstract

Given any ε ∈ (0, 1/2) and any positive integer s ≥ 2, we prove that for every prime

p ≥ max{s²(4/ε)^2s, s^{651s log log(10s)}} 

satisfying ϕ(p − 1)/(p − 1) ≤ 1/2 − ε, where ϕ(k) is the Euler function, there are s consecutive quadratic non-residues which are not primitive ischolar_mains modulo p.

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Alexandru Ciolan ¹, Florian Luca ², Pieter Moree ³

Affiliations
1 Rheinische Friedrich-Wilhelms-Universitat Bonn, Regina Pacis Weg 3, D-53113 Bonn, DE
2 School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, ZA
3 Max Planck Institut fur Mathematik, Vivatsgasse 7, D-53111 Bonn, DE

Abstract

Given a recurrent sequence U := {U_n}_n≥0 we consider the problem of counting M_U(x), the number of integers n ≤ x such that U_n = u² + nv² for some integers u, v. We will show that MU(x) ⪻ x(log x)^−0.05 for a large class of ternary sequences. Our method uses many ingredients from the proof of Alba Gonzalez and the second author [1] that M_F (x) ⪻ x(log x)^−0.06, with F the Fibonacci sequence.

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