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Math Help - pythagorean triples

  1. #1
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    pythagorean triples

    prove that if x,y,z is a Pythagorean triple, than at least one of x,y,z is divisible by 5.

    What I have so far: let x = r^2 - s^2, y = 2rs, z = r^2 + s^2
    Then try out all of the cases. If 5\nmid x \mbox{ and } 5\nmid y,\mbox{ then } 5\mid z and use congruences to show that it is true, for all of the possible values of r and s.

    Is there a more elegant way to prove this, because what I have so far is too tedious.
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  2. #2
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    Quote Originally Posted by keityo View Post
    prove that if x,y,z is a Pythagorean triple, than at least one of x,y,z is divisible by 5.

    What I have so far: let x = r^2 - s^2, y = 2rs, z = r^2 + s^2
    Then try out all of the cases. If 5\nmid x \mbox{ and } 5\nmid y,\mbox{ then } 5\mid z and use congruences to show that it is true, for all of the possible values of r and s.

    Is there a more elegant way to prove this, because what I have so far is too tedious.
    Consider the possible values of squares modulo 5, then all the possible values of x^2+y^2 modulo 5 and which of these can a square modulo 5.

    CB
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