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'Legendre's Law of Quadratic Reciprocity' ... is ...the most important general truth in the science of integral numbers which has been discovered since the time of Fermat. It has been called by Gauss 'the gem of the higher arithmetic,' and is equally remarkable whether we consider the simplicity of its enunciation, the difficulties which for a long time attended its demonstration, or the number and variety of the results which have been obtained by its means. ...e find in the 'Opuscula Analytica' of Euler... a memoir... which contains a general and very elegant theorem from which the Law of Reciprocity is immediately deducible, and which is, vice versâ, deducible from that law. But Euler... expressly observes that the theorem is undemonstrated; and this would seem to be the only place in which he mentions it in connexion with the theory of the Residues of Powers; though in other researches he has frequently developed results which are consequences of the theorem, and which relate to the linear forms of the divisors of quadratic formulae. But here also his conclusions repose on induction only; though in one memoir he seems to have imagined... that he had obtained a satisfactory demonstration. (en) |
