>On the generalized Ramanujan-Nagell equation $ x^2+(2k-1)^y = k^z $ with $ k\equiv 3 $ (mod 4)

• Main Authors: Yahui Yu, Jiayuan Hu
• Format: Article
• Language: English
• Published: AIMS Press 2021-07-01
• Series: AIMS Mathematics
• Subjects: polynomial-exponential diophantine equation; generalized ramanujan-nagell equation
• Online Access: https://www.aimspress.com/article/doi/10.3934/math.2021615?viewType=HTML

Let $ k $ be a fixed positive integer with $ k &gt; 1 $. In 2014, N. Terai ^{[<xref ref-type="bibr" rid="b6">6</xref>]} conjectured that the equation $ x^2+(2k-1)^y = k^z $ has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. This is still an unsolved problem as yet. For any positive integer $ n $, let $ Q(n) $ denote the squarefree part of $ n $. In this paper, using some elementary methods, we prove that if $ k\equiv 3 $ (mod 4) and $ Q(k-1)\ge 2.11 $ log $ k $, then the equation has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. It can thus be seen that Terai's conjecture is true for almost all positive integers $ k $ with $ k\equiv 3 $(mod 4).