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Hi I'm looking at problems on deciding if a number is square modulo a composite.

Specifically is 37 a square modulo 2018.

I can solve the problem since it's just (37/1009) = 1, but I'm wondering if dividing 2018 into 2 and 1009 can be generalized to all composites. For example if the denominator is 2024, can I calculate 2^3, 11, and 23, then multiply the Legendre symbols?

i.e. does (a/p) (a/q) = (a/pq) for p and q distinct and coprime?

Many thanks.

Bill Dubuque
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Xin Jay
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  • By Chinese remainder theorem, if $m$ and $n$ are coprime, $a$ is a square mod $mn$ iff it is a square mod $m$ and also mod $n$. [There is an extension of the Legendre symbol (Jacobi symbol and also Kronecker symbol) when the "denominator" doesn't have to be a prime, but it is no longer about just "being square modulo the denominator".] – user8268 May 12 '24 at 14:13
  • @user8268 Thanks. So one can decompose the denominator in Kronecker symbol, right? – Xin Jay May 12 '24 at 14:49
  • yes ${{{{}}}}{}$ – user8268 May 12 '24 at 16:02

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