On a diophantine equation

Abstract

In this paper the equation x2 + 32k = yn where n ≥ 3 is studied. For n = 3, it is proved that it has a solution only if k = 3K + 2 and then there is a unique solution x = 46 × 33K and y = 13 × 32K. For n > 3 theorems are proved which determine a large number of values of k and n for which this equation has no solution. It is proved that if this equation has a solution for n > 3, then n is odd and k = 2δ.k′ where δ ≥ 1, (2, δ) = 1, k′ ≡ 15 (mod 20) and all the primes divisors p of n are congruent to 11 (mod 12).

On the equation x2 + 2a · 3b = yn

“…Le [8] m is odd (see [2]). They also gave partial results in the case when m is even (see [1]) but the general solution in the case m is even was found by Luca in [11]. For any nonzero integer k, let P (k) be the largest prime dividing k. Let C 1 be any fixed positive constant.…”