For any primitive pythagorean triple $(a,b,c)$ either $a$ or $b$ must be a multiple of $3$

Question

I'm reading "Friendly Introduction to Number Theory". Now I'm working on Primitive Pythagorean Triples Exercises 2.1 (a) on P18.

We showed that in any primitive Pythagorean triple $(a, b, c)$, either $a$ or $b$ is even. Use the same sort of argument to show that either $a$ or $b$ must be a multiple of 3.

(1) $a^2 + b^2 = c^2$ with a odd, b even, a,b,c having no common factors

(2) $a^2 = c^2 - b^2 = (c-b)(c+b)$

(3) $c + b = s^2$ and $c - b = t^2$

(4) $c = \frac{(s^2 + t^2)}{2}$ and $b = \frac{(s^2 - t^2)}{2}$

(5) $a = \sqrt{(c-b)(c+b)} = st$

(6) $a = st$, $b = \frac{(s^2 - t^2)}{2}$, $c = \frac{(s^2 + t^2)}{2}$

https://www.math.brown.edu/~jhs/frintch1ch6.pdf

I have no idea how I start doing this. Can you give me a hint? I think I need to show that both the following (1) and (2) are satisfied.

$X \neq 0$

(1) $a\equiv 0\pmod 3$ and $b\equiv X\pmod 3$

(2) $b\equiv 0\pmod 3$ and $a\equiv X\pmod 3$

Answer 1

The numbers $a$ and $b$ cannot both be multiples of $3$, because then $c$ would be a multiple of $3$ and the triple would not be primitive.

And if none of them is a multiple of $3$, then both of them are of the form $3k\pm1$, for some integer $k$, from which it follows that both squares $a^2$ and $b^2$ are of the form $3k+1$. But then $c^2(=a^2+b^2)$ is of the form $3k+2$. You should be able to show that this is impossible.

Answer 2

Using https://en.m.wikipedia.org/wiki/Pythagorean_triple or https://mathcs.clarku.edu/~djoyce/java/elements/bookX/propX29.html

$a=2mn,b=m^2-n^2$ where $m,n$ are coprime integers and not both are odd

$ab=2mn(m^2-n^2)=2(m^3-m)n-2m(n^3-n)$

Now use The product of $n$ consecutive integers is divisible by $n$ factorial as $m^3-m=(m-1)m(m+1)$

Answer 3

In mod 3, $0^2=0$, $1^2=1$ and $2^2=(-1)^2=1$. Thus, if neither $a$ nor $b$ are 0 mod 3, then $a^2+b^2$ is 2 mod 3, but no number squares to 2 mod 3.

The "principal" condition doesn't really play a part in the proof.

There are also proofs that at least one number must be divisible by 4, and that at least one number must be divisible by 5.