Exercises - Pythagorean Triples

  1. Prove that if $a$, $b$, $c$ is a primitive Pythagorean triple, $a$ and $b$ can't both be even.  

  2. Prove that if $a$, $b$, $c$ is a primitive Pythagorean triple, $a$ and $b$ can't both be odd.  

  3. Prove that if $m$ and $n$ are relatively prime and $(m+n)(m-n)$ is odd, then $m+n$ and $m-n$ must also be relatively prime.
     

  4. Prove that if for integers $s$ and $t$, we know that $st$ is odd, then $s$ and $t$ must both be odd.  

  5. Prove that for odd integers $s$ and $t$, if $\displaystyle{\frac{s^2+t^2}{2}}$ and $\displaystyle{\frac{s^2-t^2}{2}}$ are relatively prime, then $s$ and $t$ must be relatively prime.

  6. Find all primitive Pythagorean triples where $c$ is less than 150.

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