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On the generalized Ramanujan-Nagell equation $ x^2+(2k-1)^y = k^z $ with $ k\equiv 3 $ (mod 4)
- Source :
- AIMS Mathematics, Vol 6, Iss 10, Pp 10596-10601 (2021)
- Publication Year :
- 2021
- Publisher :
- AIMS Press, 2021.
-
Abstract
- Let $ k $ be a fixed positive integer with $ k > 1 $. In 2014, N. Terai [6] 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).
Details
- Language :
- English
- ISSN :
- 24736988
- Volume :
- 6
- Issue :
- 10
- Database :
- Directory of Open Access Journals
- Journal :
- AIMS Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.2c1ee476e8234765a0e6a258647acc65
- Document Type :
- article
- Full Text :
- https://doi.org/10.3934/math.2021615?viewType=HTML