For ten points, identify this theorem, sometimes called "Gauss' golden theorem," which states that the congruences "x squared congruent to q mod p" and "x square congruent to p mod q" are both solvable or unsolvable unless both p and q have remainder 3 when divided by 4.
Eisenstein gave a proof of this theorem which involves writing the sine of m times x over the sine of x, where m is an odd integer, in terms of the product of sine squared of x terms minus a correction factor. A consequence of this statement is the genus theorem, which states that the Diophantine equation "x squared plus y squared equals p" is solvable for prime p if p is congruent to 1 modulo 4. This theorem is often written by setting the Legendre symbol of p over q times the Legendre symbol of q over p equal to minus one raised to to the power "p minus one times q minus one over 4," where p and q are distinct odd primes. For ten points, identify this theorem, sometimes called "Gauss' golden theorem," which states that the congruences "x squared congruent to q mod p" and "x square congruent to p mod q" are both solvable or unsolvable unless both p and q have remainder 3 when divided by 4.
Eisenstein gave a proof of this theorem which involves writing the sine of m times x over the sine of x, where m is an odd integer, in terms of the product of sine squared of x terms minus a correction factor.
quadratic reciprocity theorem [accept Gauss' golden theorem or aureum theorema before mention]