On the exponential diophantine equation (a^n-1)(b^n-1)=x^2

• Autores: László Szalay, Li Lan
• Localización: Publicationes Mathematicae Debrecen, ISSN 0033-3883, Tomus 77, Fasc. 3-4, 2010, págs. 465-470
• Idioma: inglés
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• Resumen

  • Year: 2010 Vol.: 77 Fasc.: 3-4 Title: On the exponential diophantine equation (an ¡ 1)(bn ¡ 1) = x2 Author(s): Li Lan and L¶aszl¶o Szalay Let a and b be ¯xed positive integers such that a 6= b and min(a; b) > 1. In this paper, we combine some divisibility properties of the solutions of Pell equations with elementary arguments to prove that if a ´ 2 (mod 6) and b ´ 0 (mod 3), then the title equation (an ¡1)(bn ¡1) = x2 has no positive integer solution (n; x). Moreover, we show that in case of a ´ 2 (mod 20) and b ´ 5 (mod 20), where b ¡ 1 is a full square, the only possible solution belongs to n = 1.

    Address:

    Li Lan Department of Mathematics Xi'an University of Arts & Science Xi'an 710065 P.R. China Address:

    L¶aszl¶o Szalay Institute of Mathematics and Statistics University of West Hungary Sopron Hungary