On the Diophantine Equation p^x+n^y=z^2 where p is an Odd Prime and n is a Non-Negative Integer with n≡7(mod 12) and gcd(n,p)=1

Abstract

Background and Objectives : Diophantine equations, which seek integer solutions to polynomial equations, have long been a fundamental and extensively studied topic in number theory. Among these, exponential Diophantine equations, where variables appear as exponents, are particularly challenging due to their nonlinear nature and profound connections to classical conjectures such as Catalan’s Conjecture. This conjecture, proven by Mihailescu in 2004, states that the equation x^a - y^b = 1, where a, b, x  and y  are integers with min   1  has the unique positive integer solution (x, y, a, b) = (3 ,2, 2, 3). This study focuses on the exponential Diophantine equation p^x + n^y = z²  where p  is an odd prime, and  n, x, y, z  are non-negative integers. The primary objective is to determine all possible integer solutions of this equation under various conditions, with particular emphasis on modular restrictions imposed on the parameter n.  The research investigates the interplay between the prime base  p  and the parameter n  in determining the existence of solutions. Previous works have contributed foundational insights into related problems. Notably, Nagell (1948) proved the finiteness of solutions to the Lebesgue-Nagell equation  x² + D = yⁿ, for fixed integers  and  Tijdeman (1976) extended these results using Baker’s theory of linear forms in logarithms to show the finiteness of positive integer solutions to exponential Diophantine equations of the form a^x + b^y = c,  for fixed integers a, b, c . While these studies have illuminated important aspects of the problem, a comprehensive understanding of solutions under specific modular constraints on  remains incomplete. Therefore, this research conducts a detailed analysis of the equation under the modular condition n 7(mod 12). This is achieved by employing classical number theory tools such as the theory of quadratic residues and the Legendre symbol to rigorously restrict the possible solution values. The methodology involves transforming the equation into forms amenable to modular arithmetic analysis and factorization, with the goal of establishing necessary and sufficient conditions for the existence of non-negative integer solutions. Analyzing these constraints will allow a complete characterization of all possible solution sets and demonstrate the absence of solutions outside these conditions. The results are expected to deepen the theoretical understanding of exponential Diophantine equations involving prime powers and perfect squares under modular constraints, extend classical results, and provide a framework for investigating more complex cases in future research.

Methodology : The investigation begins by transforming the equation p^x + n^y = z² where  p is an  odd prime, and n, x, y, z  are non-negative integers, into forms that are amenable to analysis using modular arithmetic and the properties of quadratic residues. A key insight involves the application of quadratic residues modulo 12, which impose stringent restrictions on the possible values of  thereby significantly  influencing the solvability of the equation. Specifically, the congruence condition  n 7(mod 12) emerges naturally from residue computations and serves as a critical criterion for filtering candidate solutions. The research further explores the relationship between the given Diophantine equation and the generalized Pell equation of the form x² - Dy²= 1 , which is known to have infinitely many integer solutions under certain conditions. By establishing this connection, the study relates the growth behavior of the original equation’s solutions to those of Pell-type equations. Techniques  derived from the theory of linear forms in logarithms, inspired by the foundational work of Tijdeman and Baker, are employed to establish explicit upper bounds on the exponents  x and  y  in terms of  p and  n. Throughout the analysis, modular constraints and the greatest common divisor condition gcd(n, p) = 1  are examined systematically. The proof strategy integrates modular arithmetic with prime factorization methods and classical  analytic number theory to derive necessary and sufficient conditions for the existence of non-negative integer solutions to the equation.

Main Results : The research shows that the Diophantine equation p^x + n^y = z² admits non-negative integer solutions only under highly restrictive circumstances. Specifically, the equation has solutions precisely when p = 3  and  gcd(n, p) = 1, with the complete solution set given by (p, n, x, y, z)  where  is a non-negative integer parameter that generates an infinite family of solutions directly related to Pell-type equations. This characterization reveals an intricate structural link between the original exponential Diophantine equation and quadratic forms. Additionally, the modular condition n 7(mod 12)  is proven to be both necessary and natural for solutions to exist. The analysis confirms that for any other values of p or n not satisfying these conditions, the equation has no non-negative integer solutions. This result not only aligns with but also extends existing theorems on the rarity of solutions to exponential Diophantine equations. It demonstrates how quadratic residue conditions can tightly constrain possible solutions, providing a deep understanding of the interplay between the arithmetic properties of  p and the modular behavior of n  Furthermore, the results establish a bridge to classical results on Pell equations by showing that the infinite familie of solutions correspond precisely to sequences generated by fundamental solutions to related Pell-type equations.

Conclusions : This study provides a complete characterization of non-negative integer solutions to the Diophantine equation p^x + n^y = z² for odd primes  p,  demonstrating that solutions exist only under the specific modular condition  n 7(mod 12)  and with p = 3  satisfying  gcd(n, p) = 1.The explicit forms of the solutions reveal a unique structure rooted in connections to Pell-type equations and quadratic residue theory. These findings deepen our understanding of exponential Diophantine equations involving prime powers and perfect squares under modular constraints. The research contributes new insight to the field by elucidating the conditions under which such equations are solvable, laying a foundation for future work on more complex equations involving higher powers or additional variables. Potential applications of the methodological advances include algorithmic number theory, cryptography, and computational mathematics, where understanding the interaction between primes, exponents, and perfect powers is crucial. By combining classical and modern techniques, this research not only extends the known results but also provides a template for studying other exponential Diophantine equations with similar structural properties, opening avenues for further exploration in both pure and applied mathematics