Explicit solutions and non-solutions for the Diophantine equation $ p^x+ q^{2y} = z^{2 n} $ involving primes $ p\not\equiv q \pmod 4 $

Abstract

Over the past decade, significant research has been conducted on the equation $ a^x+b^y = z^2 $ 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} = z^{2 n} $ 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 = z^2 $, where $ p $ and $ q $ are primes, have addressed cases where $ p = 2 $ or $ p\equiv q\pmod 4 $. In this paper, we address the case where $ p\not\equiv q\pmod 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} = z^{2n} $. We also show that the equation has no solution when $ \{2, 3\}\nsubseteq\{p, q, z\} $. In other words, we provide almost explicit solutions to $ p^x+ q^{y} = z^{2 n} $ except for the case where both $ x $ and $ y $ are odd.

Keywords

• diophantine equation
• catalan's conjecture
• legendre symbol