Thread: Show that x^5 + y^5 |= z^5 if x,y,z are integers which are not divisible by 5.

Suppose that are integers, none of which is divisible by .
Show that . Hint: work modulo .

I can only think of these congruences by Fermat's Little Theorem.


.

How can I get congruences in modulo 25 and prove the statement?

Is it proved by assuming that and then show a contradiction like the proof for in
https://en.wikipedia.org/wiki/Proof_of_Fermat's_Last_Theorem_for_specific_expone nts ?

Since:
(x+5)^5 mod 25 = x^5 (expand it out to prove)

and 1^5=1, 2^5=7, 3^5=18 and 4^5 = 24 (mod 25)

it follows that x^5, y^5 and z^5 = 1,7,18 or 24. for any x,y,z not divisible by 5.

Now we cannot add two numbers from 1,7,18 and 24 to get 1,7,18 or 24 so the equality x^5+y^5=z^5 is false.

Thank you very much. Just writing out the reason for the last step for myself. If



then

,
which means if .