Konstantine Zelator. Five Exponential Diophantine Equations and Mayhem Problem M429

Natural Sciences / Mathematics / Algebra

Submitted on: Apr 12, 2012, 05:17:18

Description: Crux Mathematicorum with Mathematical Mayhem, is a problem solving journal published by the Canadian Mathematical Society. In the March 2010 issue, the following problem was proposed: Determine all positive integers a,b, and c such that a^(b^c)=(a^b)^c. A solution by this author was published in the December 2010 issue of Crux. 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. 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.