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September 19, 2025Open Access

Solutions to Diophantine Equations Involving Primes and Perfect Squares

Key Points

Integer solutions to specific diophantine equations were successfully determined, enhancing understanding of number relationships.
The analysis used classical number theory methods including modular arithmetic and factoring strategies for comprehensive results.
Three distinct cases were examined for the first equation based on the value of prime number p, ensuring thorough exploration.
Complete solution families were produced in many instances, although some results remain partial, indicating areas for further study.

Abstract

Abstract This paper determines integer solutions ( p, m, z ), where p is a prime number, m is a perfect square, and z is a nonnegative integer, to the Diophantine equations of the form ( p + m ) x - p y = z 2 and p x - ( p + m ) y = z 2 , with x and y not both greater than 1. The analysis relies primarily on classical techniques in number theory, including factoring strategies and modular arithmetic. To structure the investigation, the first equation is thoroughly examined in three distinct cases, that is, when p = 2, when p ≡ 1 (mod 4), and when p ≡ 3 (mod 4). Results are also obtained for the second equation. Our results include complete solution families in many cases and partial findings in others.

Solutions to Diophantine Equations Involving Primes and Perfect Squares

Key Points

Abstract

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