a. Fermat’s last theorem says that for all integers n > 2,

Chapter 4, Problem 31E

(choose chapter or problem)

a. Fermat’s last theorem says that for all integers n > 2, the equation \(x^{n}+y^{n}=z^{n}\) has no positive integer solution (solution for which x, y, and z are positive integers). Prove the following: If for all prime numbers p > 2, \(x^{p}+y^{p}=z^{P}\) has no positive integer solution, then for any integer n > 2 that is not a power of 2, \(x^{n}+y^{n}=z^{n}\) has no positive integer solution.

b. Fermat proved that there are no integers x, y, and z such that \(x^{4}+y^{4}=z^{4}\). Use this result to remove the restriction in part (a) that n not be a power of 2. That is, prove that if n is a power of 2 and n > 4, then \(x^{n}+y^{n}=z^{n}\) has no positive integer solution.

Text Transcription:

x^n + y^n = z^n

x^p + y^p = z^P

x^4 + y^4 = z^4

x^n + y^n = z^n