Five Exponential Diophantine Equations and Mayhem Problem M429

TL;DR

This paper investigates and completely characterizes the positive integer solutions of five exponential Diophantine equations, extending previous work on a specific equation related to the Mayhem Problem M429.

Contribution

It provides a complete solution set for three new exponential Diophantine equations and identifies solution families for a fifth equation, advancing understanding of such equations.

Findings

Solutions for equations (2), (3), and (4) are fully characterized.

Three families of solutions are identified for equation (5).

The work extends previous results on related exponential Diophantine equations.

Abstract

Crux Mathematicorum with Mathematical Mayhem, is a problem solving journal published by the Canadian Mathematical Society. In the March 2010 issue(see reference[1]) ,the following problem was proposed:Determine all positive integers a,b, and c such that a^(b^c)=(a^b)^c; or equivalently, a^(b^c)=a^(b^c). A solution by this author was published in the December2010 issue of Crux(see reference[2]). Accordingly, all such positive integer triples are the following:The triples of the form (1,b,c); with b, c any positive integers; the triples (a,b,1); a, b positive integers, with a being at least 2; and the triples of the form (a,2,2); a being a positive integer not equal to 1.These are then the positive integer solutions to the 3-variable exponential diophantine equation, x^(y^z)=x^(yz) (1) Motivated by mayhem problem M429, in this work we investigate for more 3-variable exponential diophantine equations: x^(y^z)=x^(z^y) (2), x^(y^z)=y^(xz) (3) x^(yz)=y^(xz) (4), x^(y^z)=z^(xy) (5) We completely determine the positive integer solution sets of equations (2), (3), and (4). This is done in Theorems2,3, and4 respectively. We also find three different families of solutions to equation (5); listed in Theorem5.