Given a prime , show has no nontrivial integer solutions.

Printable View

- Aug 16th 2010, 02:08 PMchiph588@Simple Diophantine Equation
Given a prime , show has no nontrivial integer solutions.

- Aug 16th 2010, 02:21 PMUnbeatable0
Proof by infinite descent:

Suppose there axists a solution . Then since .

Therefore we also have .

Thus, after substitution and division by we get .

An analogous argument shows that and then and we get

a new solution .

We can proceed with this argument infinitely many times, which is of course absurd.

Note: This problem can be generalized to the statement that there are no

solutions to

(EDIT) where is squarefree. - Aug 17th 2010, 12:30 PMmelese
First part, to prove divisibility by the prime for and . From it's clear that and then . Dividing the equation by gives and since still divides it follows that , also . Dividing again by gives , where and are divisible by so .

and are of the forms and , respectively.Also, and are relatively prime to . Therefore, .

. If is the smallest of these, then after division we have . But then ; since and it's contradiction.

Similarly for the cases where or are the smallest powers.