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The Jacobi–Madden equation is a Diophantine equation : proposed by the physicist Lee W. Jacobi and the mathematician Daniel J. Madden in 2008.〔(Mathematicians find new solutions to an ancient puzzle )〕 The variables ''a'', ''b'', ''c'', and ''d'' can be any integers, positive, negative or 0.〔In fact, any nontrivial solution must include both a positive and negative value.〕 Jacobi and Madden showed that there are an infinitude of solutions of this equation with all variables non-zero. == History == The Jacobi–Madden equation represents a particular case of the equation : first proposed in 1772 by Leonhard Euler who conjectured that four is the minimum number (greater than one) of fourth powers of non-zero integers that can sum up to another fourth power. This conjecture, now known as Euler's sum of powers conjecture, was a natural generalization of the Fermat's Last Theorem, the latter having been proved for the fourth power by Pierre de Fermat himself. Noam Elkies was first to find an infinite series of solutions to Euler's equation with exactly one variable equal to zero, thus disproving Euler's sum of powers conjecture for the fourth power. However, until Jacobi and Madden's publication, it was not known whether there exist infinitely many solutions to Euler's equation with all variables non-zero. Only a finite number of such solutions was known.〔Jaroslaw Wroblewski (Database of solutions to the Euler's equation )〕 One of these solutions, discovered by Simcha Brudno in 1964, yielded a solution to the Jacobi–Madden equation: : 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「Jacobi–Madden equation」の詳細全文を読む スポンサード リンク
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