Quadratic reciprocity

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: Law of quadratic reciprocity — Let p and q be distinct odd prime numbers, and define the Legendre symbol as: Count the number of proofs to the law of quadratic reciprocity given thus far in this book and devise another one. In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: This law allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form x 2 ≡ a ( mod p ) {displaystyle x^{2}equiv a!!!{pmod {p}}} for p an odd prime; that is, to determine the 'perfect squares' mod p. However, this is a non-constructive result: it gives no help at all for finding a specific solution; for this, one uses quadratic residues. The theorem was conjectured by Euler and Legendre and first proved by Gauss. He refers to it as the 'fundamental theorem' in the Disquisitiones Arithmeticae and his papers, writing Privately he referred to it as the 'golden theorem.' He published six proofs, and two more were found in his posthumous papers. There are now over 240 published proofs. Since Gauss, generalizing the reciprocity law has been a leading problem in mathematics, and has been crucial to the development of much of the machinery of modern algebra, number theory, and algebraic geometry, culminating in Artin reciprocity, class field theory, and the Langlands program.