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Thread: 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 $\displaystyle 3$ is congruent to $\displaystyle 1$ modulo $\displaystyle 3$ (prove this). And no square is congurent to $\displaystyle 2$ modulo $\displaystyle 3$ (prove this also)

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

    $\displaystyle
    x^2+y^2=z^2
    $

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

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

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

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

    CB
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