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Can you help me understand what my friend was saying to me the other day? He's really smart (a math savant AND a huge fan of those cryptic crossword things), so I'm trying to spend time with him and have his genius rub off on me, but he can be really pretentious and hard to understand.

He was writing in his math notebook, and when I looked over his shoulder, I saw the following equation:

$(\frac{a}{m})(\frac{b}{m}) = (\frac{ab}{m})$

I told him it didn't look right, but he only stopped for a second to consider it, and he said "Yes it is" without even lifting his head!

I thought about it for some time, and eventually said, "Oh, I understand. $m$ has to be equal to 1."

"No, actually, $m$ can't be equal to 1," he quickly responded. Then, after a second of writing, he stopped again. "Well, actually, I guess that $m$ has to be synonymous to the number 1, in a sense."

I was bewildered. I looked at the equation again, desperate to make sense of it or what he just told me. Finally, I gave up and said, "I don't believe you."

He stopped again to consider it, and eventually gave me a pretentious, proud smirk. He pointed at one side of the equation and said, "Well, if you avoid the repetition at the end, you'll find something that's hard to tell if it's true or false. But it's also the key."

Can you tell me what my friend meant by any of that, and how his equation could possibly be true?

Note: (unrelated to puzzle): I read a lot of questions on this and other stack exchanges, but this is my first time interacting. Please let me know if there's some aspect of the etiquette I'm missing!

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    $\begingroup$ Welcome to Puzzling Stack Exchange. Good job on your first puzzle - it looks great. I don't have an answer yet, but I'm thinking about it... $\endgroup$
    – Earlien
    Mar 15 at 10:26
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Perhaps the equation refers to the

Legendre symbol.

In particular,

$\left( \frac a p\right)$ returns 1 if $a$ is a quadratic residue mod $p$, -1 if not, and 0 in the edge case where it's actually a multiple of $p$.

If that is the case, the equation is indeed true:

by the multiplicativity of the Legendre symbol.

Here $m$ has to be

a "prime", which can be a synonym for "main" or "the #1".

"Well, if you avoid the repetition at the end, you'll find something that's hard to tell if it's true or false. But it's also the key."

If we ignore the "re" (repetition) at the end, "Legendre" reads "legend". It's usually hard to verify whether legends are true or false, but "legend" can also mean "key" (as in a map).

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