Journal of the Ramanujan Mathematical Society
Volume 32, Issue 2, June 2017 pp. 165–183.
Counting terms Un of third order linear recurrences with Un = u2 + nv2
Authors:
Alexandru Ciolan, Florian Luca and Pieter Moree
Author institution:Rheinische Friedrich-Wilhelms-Universität Bonn, Regina Pacis Weg 3, D-53113 Bonn, Germany
Summary:
Given a recurrent sequence U := {Un}n ≥ 0 we consider
the problem of counting MU(x), the number of
integers n ≤ x such that Un = 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 González and the second
author [1] that MF(x) << x(log x)-0.06,
with F the Fibonacci sequence.
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