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 Definition/Summary A number n is a quadratic residue mod m if there exists some number a which, squared mod m, gives n.

 Equations Definition of the Legendre symbol, for any number a and for any odd prime p: $$\left(\frac ap\right)=\begin{cases} 0&p|a\\ 1&\exists n:n^2\equiv a\pmod p\\ -1&\nexists n:n^2\equiv a\pmod p \end{cases}$$ The Legendre symbol is multiplicative: $$\left(\frac{ab}{p}\right)=\left(\frac ap\right)\left(\frac bp\right)$$ The Law of Quadratic Reciprocity, for any odd primes p and q: $$\left(\frac qp\right)=(-1)^{(p-1)(q-1)/4}\left(\frac pq\right)$$

 Scientists Euler, Gauss, Legendre, Jacobi, Hilbert