On a diophantine equation

Abstract

In this paper the equation x 2 + 3 2 k = y n where n ≥ 3 is studied. For n = 3, it is proved that it has a solution only if k = 3 K + 2 and then there is a unique solution x = 46 × 3 3 K and y = 13 × 3 2 K . 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 = 2 δ . 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).