be integers. For every prime, p, dividing abc if the congruence has a nontrivial solution, then so does: Zeta function formulation As mentioned in the article on Dedekind zeta functions, quadratic reciprocity is equivalent to the zeta function of a quadratic field being the product of the Riemann zeta function and a certain Dirichlet L-function Jacobi symbol The Jacobi symbol is a generalization of the Legendre symbol; the main difference is that the bottom number has to be positive and odd, but does not have to be prime. If it is prime, the two symbols agree. It obeys the same rules of manipulation as the Legendre symbol. In particular and if both numbers are positive and odd (this is sometimes called "Jacobi's reciprocity law"): However, if the Jacobi symbol is 1 but the denominator is not a prime, it does not necessarily follow that the numerator is a quadratic residue of the denominator. Gauss's cases 9) - 14) above can be expressed in terms of Jacobi symbols: and since p is prime the left hand side is a Legendre symbol, and we know whether M is a residue modulo p or not. The formulas listed in the preceding section are true for Jacobi symbols as long as the symbols are defined. Euler's formula may be written Example. 2 is a residue modulo the primes 7, 23 and 31: But 2 is not a quadratic residue modulo 5, so it can't be one modulo 15. This is related to the problem Legendre had: if then a is a non-residue modulo every prime in the arithmetic progression m + 4a, m + 8a, ..., if there are any primes in this series, but that wasn't proved until decades after Legendre. Eisenstein's formula requires relative primality conditions (which are true if the numbers are prime) Let be positive odd integers such that: Then Hilbert symbol The quadratic reciprocity law can be formulated in terms of the Hilbert symbol where a and b are any two nonzero rational numbers and v runs over all the non-trivial absolute values of the rationals (the Archimedean one and the p-adic absolute values for primes p). The Hilbert symbol is 1 or −1. It is defined to be 1 if and only if the equation has a solution in the completion of the rationals at v other than . The Hilbert reciprocity law states that , for fixed