I Properties of the legs of a right triangle

[e2m2a]

TL;DR Summary

Is it true that there can be no more than one leg of a right triangle that is a perfect power of a number?

I want to know if a right triangle can only have one leg that is a perfect power of a number. Another words is it impossible for a right triangle to have two legs that are numbers that are raised to the same perfect power? Can somebody answer this question and show me the proof?

 

[jedishrfu]

I'm not sure I understand your question. Is it related to Pythagorean triples where only one element of the triple is some power of a number as in 4, 8, 16, 64 or 6, 9 as shown in the red hilighted triples below.

https://en.wikipedia.org/wiki/Pythagorean_triple

There are 16 primitive Pythagorean triples of numbers up to 100:

(3, 4, 5)	(5, 12, 13)	(8, 15, 17)	(7, 24, 25)
(20, 21, 29)	(12, 35, 37)	(9, 40, 41)	(28, 45, 53)
(11, 60, 61)	(16, 63, 65)	(33, 56, 65)	(48, 55, 73)
(13, 84, 85)	(36, 77, 85)	(39, 80, 89)	(65, 72, 97)

 

[Delta2]

I am not sure I understand your question either, but anyway here is my take:
We can construct right triangles with the legs to be anything we want, but then the hypotenuse won't be anything we want, it would be such that it is equal to the square root of the sum of squares of the legs.

 

[pbuk]

I think the question is "do there exist integers a, b, c and n such that $a^{2n} + b^{2n} = c^2$.

This is clearly related to Fermat's Last Theorem (FLT - although that should not be confused with Faster than Light Travel which we only talk about in the science fiction topic ).