Journal of the Ramanujan Mathematical Society

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.

Full Text