= Jacobi reciprocity law
{c}
{title2=$(a/b)(b/a)=(-1)^{(a-1)(b-1)/4}$}
= Jacobi reciprocity
{c}
{synonym}
For coprime positive odd integers $a,b$, the <Jacobi symbol> satisfies $(a/b)(b/a)=(-1)^{(a-1)(b-1)/4}$. It follows by factoring $a$ and $b$ into primes and multiplying the <quadratic reciprocity> for each prime pair; the parity of the sums of prime half-differences agrees with $(a-1)/2$ and $(b-1)/2$. A <Jacobi symbol> equal to one does not imply that its numerator is a <quadratic residue> for a composite denominator.
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