Odd Pythagorian triplets

Question

How many Pythagorean triplets $\{a,b,c\}$ exist, where $a,b,c$ are all odd?

As far as I know there are no such triplets. $\{3, 4, 5\}; \{5,12,13\} ; \{7,24,25\}$ and its multiples are examples.

Is there any explanation on why all the integers forming a triplet are not all odd or all even?

Answer 1

Suppose the numbers are $a,b,c$. Suppose $a^2+b^2=c^2$. The order doesn't matter anyway. Since $a$ and $b$ are odd, $a^2+b^2$ will be even. But, $c^2$ is odd. Hence, no such triplet can exist.

Answer 2

Think about

$a^2b^2(a^2+b^2)$ mod $2$

$a^2b^2(a^2+b^2+2ab)$ mod $2$

$a^2b^2(a+b)^2$ mod $2$

If both $a$ and $b$ are such that $a \equiv 1 \text{ mod } 2$ and $b \equiv 1 \text{ mod } 2$ then $a+b \equiv 0 \text{ mod } 2$ hence abc is divisible by 2.

Answer 3

Use that $$a^2+b^2=c^2\iff(\lambda^2-\mu^2)^2+(2\lambda\mu)^2=(\lambda^2+\mu^2)^2$$