I Properties of the legs of a right triangle

AI Thread Summary
A right triangle can have one leg that is a perfect power, but it is not guaranteed that both legs can be perfect powers raised to the same exponent. The discussion references Pythagorean triples, emphasizing that while various combinations exist, the relationship between the legs and the hypotenuse must satisfy the equation a² + b² = c². The question posed relates to whether integers can exist such that a^(2n) + b^(2n) = c², which connects to Fermat's Last Theorem. The conversation highlights the complexity of constructing right triangles with specific properties regarding their legs. Ultimately, the existence of such triangles depends on the values chosen for the legs and their mathematical relationships.
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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?
 
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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)
 
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.
 
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 :biggrin:).
 
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