prove that 5^x = m^2 + n^2 for any x

Question

prove that 5^x = m^2 + n^2 for any x 

please reply asap

Answer 1

The right side has the form of the right side of the Pythagorean Theorem, so we know that it represents some positive number, assuming that m and n are Real numbers.

 

Any positive Real number can be represented as 5^x -- FOR SOME SPECIFIC x (which can be found using logarithms).

 

So the converse must be true; given any Real x, it can be written in the form of the right side, although one might be discomfited trying to find specific m and n values.