A note on the Diophantine equation $x^{2} + q^{m} = c^{2n}$

Abstract

Let q be an odd prime. Let c > 1 and t be positive integers such that q t þ 1 ¼ 2c 2 . Using elementary method and a result due to Ljunggren concerning the Diophantine equation x n À1 xÀ1 ¼ y 2 , we show that the Diophantine equation x 2 þ q m ¼ c 2n has the only positive integer solution ðx; m; nÞ ¼ ðc 2 À 1; t; 2Þ. As applications of this result some new results on the Diophantine equation x 2 þ q m ¼ c n and the Diophantine equation x 2 þ ð2c À 1Þ m ¼ c n are obtained. In particular, we prove that Terai's conjecture is true for c ¼ 12; 24. Combining this result with Terai's results we conclude that Terai's conjecture is true for 2 c 30.

A Note on the Diophantine Equation

“…Using these results, together with results of Ljunggren [5], Zhu [7] and Arif-Abu Muriefah [1], Terai showed that, apart from c = 12, 24, his conjecture holds for 2 ≤ c ≤ 30. The cases c = 12, 24 have been treated in [4]. In this paper, we show that Terai's conjecture is true under a wider range of conditions on c and 2c − 1.…”

“…and[4], we may suppose that 31 ≤ c ≤ 499 with c ≡ 3 (mod 4).For c = p 2s+1 , where p is a prime, p ≡ 3 (mod 4), s ≥ 0 and 31 ≤ p 2s+1 ≤ 499, that is, c ∈ {31,43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, …”

A modular approach to the generalized Ramanujan-Nagell equation

“…The above conjecture has been verified in some special cases (see [1,[10][11][12][13]15], [23] and [26]). But, in general, it is far from solved problem.…”

Sums of two $S$-units via Frey-Hellegouarch curves

“…For each set S = {2, 3, p}, we have tabulated these solutions in the file http://www.math.ubc.ca/~bennett/2-3-p.pdf. By quick examination of this table for p = 23 and p = 47, we immediately deduce the following result about equations that remained unsolved in Proposition 3.3 of Terai [24] (but have been recently solved via rather different methods by Deng [11]).…”

“…We now deal with equation (2) when n = 3 using a (p, p, 3) Frey-Hellegouarch curve approach. The corresponding equations to treat are of the shape (11) x + y = wz 3 where x, y and w are pairwise coprime S-units. We may assume, without loss of generality, that w is cubefree and positive and that we have x ≡ 0 (mod 3) and y ≡ 2 (mod 3).…”