A note on the polynomial-exponential Diophantine equation (an − 1)(bn − 1) = x2

Yasutsugu Fujita and Maohua Le
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 3, Pages 123–129
DOI: 10.7546/nntdm.2021.27.3.123-129
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Authors and affiliations

Yasutsugu Fujita
Department of Mathematics, College of Industrial Technology, Nihon University
2-11-1 Shin-ei, Narashino, Chiba, Japan

Maohua Le
Institute of Mathematics, Lingnan Normal College
Zhangjiang, Guangdong, 524048 China

Abstract

For any positive integer t, let ord_2 t denote the order of 2 in the factorization of t. Let a,\,b be two distinct fixed positive integers with \min\{a,b\}>1. In this paper, using some elementary number theory methods, the existence of positive integer solutions (x,n) of the polynomial-exponential Diophantine equation (*) (a^n-1)(b^n-1)=x^2 with n>2 is discussed. We prove that if \{a,b\}\ne \{13,239\} and ord_2(a^2-1)\ne ord_2(b^2-1), then (*) has no solutions (x,n) with 2\mid n. Thus it can be seen that if \{a,b\}\equiv \{3,7\},\{3,15\},\{7,11\},\{7,15\} or \{11,15\} \pmod{16}, where \{a,b\} \equiv \{a_0,b_0\} \pmod{16} means either a \equiv a_0 \pmod{16} and b \equiv b_0\pmod{16} or a\equiv b_0 \pmod{16} and b\equiv a_0 \pmod{16}, then (*) has no solutions (x,n).

Keywords

  • Polynomial-exponential Diophantine equation
  • Pell’s equation

2020 Mathematics Subject Classification

  • 11D61

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Cite this paper

Fujita, Y. & Le, M. (2021). A note on the polynomial-exponential Diophantine equation (an − 1)(bn − 1) = x2. Notes on Number Theory and Discrete Mathematics, 27(3), 123-129, doi: 10.7546/nntdm.2021.27.3.123-129.

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