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I have a quick question regarding modular arithmetic.

If I have a Pythagorean Triple $(a, b, c)$, is it possible to consider this equation $\mod{c}$. That is to say,

Is the implication

$$a^2 + b^2 = c^2 \implies (a \mod{c})^2 + (b \mod{c})^2 \equiv 0\mod{c}$$

true?

I attempted to prove this (rather naïvely) by remembering that

\begin{align} a + cx &= a \mod{c}\\ b + cy &= b \mod{c}\\ cz &= 0 \mod{c} \end{align}

for some $x, y, z \in \Bbb{Z}.$ Squaring these expressions gives me

$$a^2 + b^2 + 2cx(a + b) + c^2(x^2 + y^2) = cz$$

and hence that $$a^2 + b^2 \equiv 0 \mod{c}.$$

Have I made some fundamental error or is this a valid method?

Edit: Changed $c^2z^2$ to $cz$.

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  • $\begingroup$ You are right: From $a^2+b^2=c^2$, we caan conclude immediately $a^2+b^2\equiv 0\pmod{c}$. $\endgroup$ – André Nicolas Apr 18 '16 at 18:23
  • $\begingroup$ Oh of course, unnecessary proof. In that case, could it then be easier to check if a triple is pythagorean by looking at $a^2$ and $b^2$ modulo $c$ ? $\endgroup$ – ÍgjøgnumMeg Apr 18 '16 at 18:27
  • $\begingroup$ If $a^2+b^2\equiv 0\pmod{c}$, it does not necessarily follow that $a^2+b^2=c^2$. But certainly if $a^2+b^2\not\equiv 0\pmod{c}$, then $(a,b,c)$ is not a triple. I do not know whether the test you suggest might speed up things in some contexts. $\endgroup$ – André Nicolas Apr 18 '16 at 18:36
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It has to be true due to a mod b=a if a

Proof: Since a and b are smaller than c, $((a\mod c)^2+(b\mod c)^2)\mod c=(a^2+b^2)\mod c=c^2\mod c=0$

Q.E.D.

And yes, your method is valid too, it is just a longer version of the method above

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