On a class of Lebesgue-Ljunggren-Nagell type equations

Abstract

Text. Given odd, coprime integers a, b (a > 0), we consider the Diophantine equation ax(2) + b(2l) = 4y(n), x, y is an element of Z, l is an element of N, n odd prime, gcd(x, y) = 1. We completely solve the above Diophantine equation for a is an element of {7, 11, 19, 43, 67, 163}, and b a power of an odd prime, under the conditions 2(n-1)b(l) not equivalent to +/- 1(mod a) and gcd (n, b) = 1. For other square-free integers a > 3 and b a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers x, y with (gcd(x, y) = 1), l is an element of N and all odd primes n > 3, satisfying 2(n-1)b(l) not equivalent to +/- 1(mod a), gcd(n, b) = 1, and gcd(n, h(-a)) = 1, where h(-a) denotes the class number of the imaginary quadratic field Q(root-a). Video. For a video summary of this paper, please visit https://youtu.be/Q0peJ2GmqeM.