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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. There are a number of equivalent statements of the theorem. One version of the law states that : for ''p'' and ''q'' odd prime numbers, and denoting the Legendre symbol. This law, combined with the properties of the Legendre symbol, means that any Legendre symbol can be calculated. This makes it possible to determine, for any quadratic equation, , where ''p'' is an odd prime, if it has a solution. However, it does not provide any help at all for actually ''finding'' the solution. The solution can be found using quadratic residues. The theorem was conjectured by Euler and Legendre and first proved by Gauss.〔Gauss, DA § 4, arts 107–150〕 He refers to it as the "fundamental theorem" in the ''Disquisitiones Arithmeticae'' and his papers, writing :''The fundamental theorem must certainly be regarded as one of the most elegant of its type.'' (Art. 151) Privately he referred to it as the "golden theorem."〔E.g. in his mathematical diary entry for April 8, 1796 (the date he first proved quadratic reciprocity). See (facsimile page from Felix Klein's ''Development of Mathematics in the 19th century'' )〕 He published six proofs, and two more were found in his posthumous papers. There are now over 200 published proofs.〔See F. Lemmermeyer's chronology and bibliography of proofs in the external references〕 The first section of this article gives a special case of quadratic reciprocity that is representative of the general case. The second section gives the formulations of quadratic reciprocity found by Legendre and Gauss. ==Motivating example== Consider the polynomial ''f''(''n'') = ''n''2 − 5 and its values for ''n'' = 1, 2, 3, 4, ... The prime factorizations of these values are given as follows: A striking feature of the data is that with the exceptions of 2 and 5, ''the prime numbers that appear as factors are precisely those with final digit 1 or 9.'' Another way of phrasing this is that the primes ''p'' for which there exists an ''n'' such that ''n''2 ≡ 5 (mod ''p'') are precisely 2, 5, and those primes ''p'' that are ≡ 1 or 4 (mod 5). The law of quadratic reciprocity gives a similar characterization of prime divisors of ''f''(''n'') = ''n''2 − ''c'' for any integer ''c''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quadratic reciprocity」の詳細全文を読む スポンサード リンク
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