What type of study is this?

This is a Mathematical Induction study (also classified as: Observational).

September 10, 2025Open Access

On the Diophantine Equation pˣ + (p + 5k) ʸ = z²

Key Points

Nonnegative integer solutions are explored for the equation $p^x + (p + 5k)^y = z^2$.
Results obtained focus on the cases of $p=2$ and when $p$ and $p+5k$ are prime pairs.
Modular arithmetic and mathematical induction are key methods employed in this exploration.
The investigation is confined to solutions where $x$ and $y$ are not both greater than 1.

Abstract

With the use of modular arithmetic and other fundamental number theoretic methods, as well as the concepts of floor function and the principle of mathematical induction, this study searches for possible nonnegative integer solutions of exponential Diophantine equations of the form pˣ + (p+5k) ʸ = z², where k N. Results are obtained for the following cases: itemize a) when p =2; or b) when p and p + 5k are prime pairs. In addition, the study is limited only to solutions where x and y are not both greater than 1.

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Cite This Study

Bacani et al. (Fri,) studied this question.

synapsesocial.com/papers/68c1a77a54b1d3bfb60e0c48 https://doi.org/https://doi.org/10.29020/nybg.ejpam.v18i3.6593