Divisibility on the sequence of perfect squares minus one: The gap principle

JOURNAL OF NUMBER THEORY

卷:184

Article

Tsz Ho Chan¹  Choi, Stephen²  Lam, Peter Cho-Ho² 

[1] Univ Memphis, Dept Math Sci, Memphis, TN 38152 USA

[2] Simon Fraser Univ, Dept Math, Burnaby, BC V5A 1S6, Canada

关键词: Gap principle;    Pell's equations;    Quadratic congruence;   

DOI  :  10.1016/j.jnt.2017.08.032

来源: Elsevier

【 摘 要 】

In this paper, we consider a gap principle when a(2) - 1 vertical bar b(2) - 1 vertical bar c(2) - 1 with 1 < a < b < c. As a byproduct, we are led to determine the complete set of pairs of positive integers 1 <= u <= v <= x such that u vertical bar v(2) - 1 and v vertical bar u(2) - 1 and the diophantine equation u(2) + v(2) 1 = muv. We also generalize our main theorems to the polynomial f (n) = A(n + B)(2) + C. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free