Quadratic reciprocity
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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
Gauss published the first and second proofs of the law of quadratic reciprocity on arts 125–146 and 262 of Disquisitiones Arithmeticae in 1801.
Part of Article 131 in the first edition (1801) of the Disquisitiones, listing the 8 cases of quadratic reciprocity
Jacobi symbol
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Jacobi symbol for various k and n. Only 0 ≤ k < n are shown, since due to rule below any other k can be reduced modulo n. Quadratic residues are highlighted in yellow — note that no entry with a Jacob
Quadratic reciprocity
…these, 3, 7, and 11 ≡ 3 (mod 4), so m ≡ 3 (mod 4). 5×7×11 R 3; 3×7×11 R 5; 3×5×11 R 7; and 3×5×7 N 11, so there are an odd number of nonresidues. 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…
Carl Gustav Jacob Jacobi who introduced the symbol.