Properties of the legs of a right triangle

In summary, the conversation discusses the possibility of a right triangle having two legs that are perfect powers of a number. It references Pythagorean triples and Fermat's Last Theorem as potential factors in determining whether this is possible. The conclusion is that while we can construct right triangles with any desired legs, it is unlikely that there exist integers that satisfy the equation mentioned in the question.
  • #1
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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?
 
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  • #2
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)
 
  • #3
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.
 
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  • #4
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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Related to Properties of the legs of a right triangle

1. What are the three sides of a right triangle called?

The three sides of a right triangle are called the hypotenuse, the opposite side, and the adjacent side.

2. How are the sides of a right triangle related?

The sides of a right triangle are related through the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

3. What is the relationship between the angles and sides of a right triangle?

The angles of a right triangle are always 90 degrees and 45 degrees, with the hypotenuse opposite the 90 degree angle. The sides of a right triangle are also related through trigonometric functions such as sine, cosine, and tangent.

4. How can the properties of a right triangle be used in real-life applications?

The properties of a right triangle have many real-life applications, such as in construction and engineering for calculating distances and angles, in navigation for determining direction and location, and in physics for analyzing forces and motion.

5. How can the properties of a right triangle be used to find missing sides or angles?

The properties of a right triangle, such as the Pythagorean theorem and trigonometric functions, can be used to solve for missing sides or angles. By setting up and solving equations, the unknown values can be determined using the known values of the triangle.

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