can it be proven that the pathagorean theorum only works with an exponent of 2?

Answer 1

I can see two different interpretations of your question.

Interpretation 1: You are asking that if you have a right triangle with sides of length a and b, and hypotenuse of length c, then a^2 + b^2 = c^2. No other powers work.

Interpretation 2: You are asking if there are solutions to a^n + b^n = c^n for n other than 2. In this case, for n > 2 there are no nontrivial RATIONAL solutions. This is the content of what is known as "Fermat's Last Theorem".

Answer 2

say you have a triangle whose hypotenuse is 8 and the other two legs are 2 each. An exponent of 0 in the pythagorean theorum concludes that 1+1 (2) = 1, which is not true. Two does work.
3: 8+8= 512, which obviously does not work.
-1: 1/2+1/2=1/8, does not work.
-2: 1/4+1/4=1/64, does not work.
Thus you can assume that this pattern will continue and that the answer is yes, it only works with an exponent of two.

Answer 3

Mathematicians tried to prove Fermat's Last Theorem for hundreds of years. In 1993 a proof of Fermat's Last Theorem was announced by Andrew Wiles of Princeton University but later a mistake was found in his proof. After more work he and another mathematician, Richard Taylor, corrected the mistake in the proof. While the theorem is simple to understand, the proof is over 200 pages long and uses advanced mathematical techniques than even most mathematicians don't understand.

http://www-history.mcs.st-andrews.ac.uk/~history/HistTopics/Fermat's_last_theorem.html
http://en.wikipedia.org/wiki/Fermat's_last_theorem

Answer 4

No, it definately works with an exponent of 0.

Answer 5

A better question is what is the relationship for geometric objects in higher exponents.

Answer 6

For non-zero integers and equal powers greater than 3 it is not solvable.