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Math Help - Pythagorean Triples

  1. #1
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    Pythagorean Triples

    Let x,y,z be a primitive Pythagorean Triple with y even.

    a)Prove that exactly one of x and y is divisible by 3. (Hint: Proof by contradiction)

    b)Prove that exactly one of x and y is divisible by 4.
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  2. #2
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    Quote Originally Posted by mndi1105 View Post
    Let x,y,z be a primitive Pythagorean Triple with y even.

    a)Prove that exactly one of x and y is divisible by 3. (Hint: Proof by contradiction)
    The square of a number not divisible by 3 is congruent to 1 modulo 3 (prove this). And no square is congurent to 2 modulo 3 (prove this also)

    Suppose x and y are not divisible by 3, then the left hand side of:

     <br />
x^2+y^2=z^2<br />

    is congruent to 2 modulo 3 while the right hand side cannot be congurent to 2 modulo 3, a contradiction, hence at least one of x and y is divisible by 3.

    If both x and y are divisible 3 then the left hand side is divisible by 3 but the right hand side cannot be divisivle by 3 because \{x,y,z\} is a primitive Pythagorean triple and so x,y,x share no common factor. This is a contradiction, hence both x and y cannot be divisible by 3.

    Hence exactly one of x,y is divisible by 3.

    (note we have not used the fact that y is even)

    CB
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