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Math Help - Show that x^5 + y^5 |= z^5 if x,y,z are integers which are not divisible by 5.

  1. #1
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    Show that x^5 + y^5 |= z^5 if x,y,z are integers which are not divisible by 5.

    Suppose that x,y,z are integers, none of which is divisible by 5.
    Show that x^5 + y^5 \neq z^5. Hint: work modulo 25.

    I can only think of these congruences by Fermat's Little Theorem.
    x^5 \equiv x (\mod 5)
    y^5 \equiv y (\mod 5)
    z^5 \equiv z (\mod 5).

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

    Is it proved by assuming that x^5 + y^5 = z^5 and then show a contradiction like the proof for n = 5 in
    https://en.wikipedia.org/wiki/Proof_of_Fermat's_Last_Theorem_for_specific_expone nts ?
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  2. #2
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    Re: Show that x^5 + y^5 |= z^5 if x,y,z are integers which are not divisible by 5.

    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.
    Thanks from math2011
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  3. #3
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    Re: Show that x^5 + y^5 |= z^5 if x,y,z are integers which are not divisible by 5.

    Thank you very much. Just writing out the reason for the last step for myself. If
    x^5 \equiv a \pmod{25}
    y^5 \equiv b \pmod{25}
    z^5 \equiv c \pmod{25}
    then
    x^5 + y^5 \equiv a + b \pmod{25}
    z^5 \equiv c \pmod{25},
    which means a + b = c if x^5 + y^5 = z^5.
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