Abstract

Over the past decade, significant research has been conducted on the equation a^x + b^y = z² under various conditions imposed on a and b or on x and y. Most studies focus on conditions where the equation has no solution, while some explore cases with infinitely many solutions, often considering scenarios where x or y is even. Motivated by this line of inquiry, we have been inspired to investigate and analyze equations of the form p^x + q^2y = z²ⁿ for two distinct primes p and q, and to present explicit forms of their solutions (p, x, q, y, z, n). Recent studies on the exponential Diophantine equation p^x + q^y = z², where p and q are primes, have addressed cases where p = 2 or p ≡ q (mod 4). In this paper, we address the case where p ≢ q (mod 4) and y is even. In addition, we explore special cases where z is the prime and provide the complete set of solutions for p^x + q^2y = z²ⁿ. We also show that the equation has no solution when {2, 3} ⊈ {p, q, z}. In other words, we provide almost explicit solutions to p^x + q^y = z²ⁿ except for the case where both x and y are odd.