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Why does the Legendre symbol, $(\frac{a}{p})$, apply only to odd primes $p$; and thusly the Jacobi symbol only to odd numbers? Are there any similar identities that hold for $p=2$?

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$\mod 2$, $1$ is a quadratic residue trivially.

The Legendre symbol applies only to odd primes since its explicit definition does not make sense for $p=2$:

$(\frac{a}{p}) \equiv a^{(p-1)/2}$.

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