Journal of the Ramanujan Mathematical Society

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Counting Terms Un of Third Order Linear Recurrences with Un = U2 + nv2


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

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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 Gonzalez and the second author [1] that MF (x) ⪻ x(log x)−0.06, with F the Fibonacci sequence.
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  • Counting Terms Un of Third Order Linear Recurrences with Un = U2 + nv2

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Authors

Alexandru Ciolan
Rheinische Friedrich-Wilhelms-Universitat Bonn, Regina Pacis Weg 3, D-53113 Bonn, Germany
Florian Luca
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
Pieter Moree
Max Planck Institut fur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

Abstract


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 Gonzalez and the second author [1] that MF (x) ⪻ x(log x)−0.06, with F the Fibonacci sequence.