27 Functions of Number TheoryMultiplicative Number Theory27.8 Dirichlet Characters27.10 Periodic Number-Theoretic Functions

For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. If $p$ divides $n$, then the value of $(n|p)$ is $0$. If $p$ does not divide $n$, then $(n|p)$ has the value $1$ when the quadratic congruence ${x}^{2}\equiv n\phantom{\rule{veryverythickmathspace}{0ex}}(modp)$ has a solution, and the value $-1$ when this congruence has no solution. The Legendre symbol $(n|p)$, as a function of $n$, is a Dirichlet character (mod $p$). It is sometimes written as $(\frac{n}{p})$. Special values include:

27.9.1 | $(-1|p)$ | $={(-1)}^{(p-1)/2},$ | ||

27.9.2 | $(2|p)$ | $={(-1)}^{({p}^{2}-1)/8}.$ | ||

If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that

27.9.3 | $$(p|q)(q|p)={(-1)}^{(p-1)(q-1)/4}.$$ | ||

If an odd integer $P$ has prime factorization $P={\prod}_{r=1}^{\nu \left(n\right)}{p}_{r}^{{a}_{r}}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod}_{r=1}^{\nu \left(n\right)}{(n|{p}_{r})}^{{a}_{r}}$, with $(n|1)=1$. The Jacobi symbol $(n|P)$ is a Dirichlet character (mod $P$). Both (27.9.1) and (27.9.2) are valid with $p$ replaced by $P$; the reciprocity law (27.9.3) holds if $p,q$ are replaced by any two relatively prime odd integers $P,Q$.