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ON THE NUMBER OF DIVISORS OF $n^{2}-1$

Part of: Elementary number theory Multiplicative number theory

Published online by Cambridge University Press:  02 October 2015

ADRIAN W. DUDEK
ADRIAN W. DUDEK*
Affiliation:
Mathematical Sciences Institute, The Australian National University, Canberra, Australia email adrian.dudek@anu.edu.au
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Abstract

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We prove an asymptotic formula for the sum $\sum _{n\leq N}d(n^{2}-1)$, where $d(n)$ denotes the number of divisors of $n$. During the course of our proof, we also furnish an asymptotic formula for the sum $\sum _{d\leq N}g(d)$, where $g(d)$ denotes the number of solutions $x$ in $\mathbb{Z}_{d}$ to the equation $x^{2}\equiv 1~(\text{mod}~d)$.

Keywords

divisor sumasymptotic estimatearithmetic functionsDiophantine quintuples

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 11A07: Congruences; primitive roots; residue systems

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 2 , April 2016 , pp. 194 - 198
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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