On the number of solutions of the generalized Ramanujan-Nagell equation

A remark on the Diophantine equation (x^3-1)/(x-1)=(y^n-1)/(y-1)

“…It has been conjectured that the equation (2) has only a finite number of solutions, even that has only two solutions (x, y, m, n) = (5, 2, 3, 5), (90,2,3,13). This is rather a difficult question.…”

“…If we prove that (3) has no exceptional solutions, then the conjecture is true under the condition m = 3. Le [12] proved that (3) has no exceptional solution with ω(y) > 1, where ω(a) denote the number of distinct prime divisors of a (the reference [12] contains an error, one can refer to [2] for a correct version). Nesterenko and Shorey [16] proved that any exceptional solution of (3) with 2 ∤ n must be n ≥ 25.…”

On the Diophantine equation (xm-1)/(x-1) = (yn-1)/(y-1)

“…Modulo 8, we see that u is odd, hence (5 (u−1)/2 , v) is a solution of 5X 2 + 3 = 2 k . By Theorem 1 of [5], this equation has only two solutions, namely (X, k) = (1, 3) and (5, 7).…”

“…Finally, it remains us to treat the pair (3,5), hence the equation 2 · 3 u − 5 v = 1. Modulo 3, we see that v is odd, thus (5 (v−1)/2 , u) is a solution of 1 + 5X 2 = 2 · 3 k .…”

The Diophantine equation 2𝑥²+1=3ⁿ

“…In the proof of Theorem 2 of [2], Bugeaud and Shorey used Le's result N (D 1 , 1, p) ≤ 2 to claim that N (2, 1, 3) = 2, by giving the solutions (x, n) = (1, 1) and (2,2). (See also the remarks on page 59 of [2]. )…”