E716  Resolutio formulae Diophanteae ab(maa + nbb) = cd(mcc + ndd) per numeros rationales
(The resolution of the Diophantine formula ab(maa+nbb) = cd(mcc+ndd) by rational numbers)
Summary:
In solving A^{4} + B^{4} = C^{4} + D^{4}, it was reduced to solving
in integers ab(aa + bb) = cd(cc + dd). This paper generalizes that. It opens with
a citation of Fermat's Last Theorem, with the remark that Fermat's proof has been lost. Then it makes the conjecture that two
cubes can't sum to be a cube, three fourth powers a 4^{th} power, or, in general, n  1 sums of
n^{th} powers can't be an n^{th} power. [This conjecture has been shown to be false.]
According to the records it was presented to the St.
Petersburg Academy on December 17, 1778.
Publication:

Originally published in Nova Acta Academiae Scientarum Imperialis Petropolitinae 13, 1802, pp. 4563

Opera Omnia: Series 1, Volume 4, pp. 329  351
 Reprinted in Commentat. arithm. 2, 1849, pp. 281293 [E716a]
Documents Available:
 Original Publication: E716
 The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E716 include:
 Lander LJ., “Geometric aspects of Diophantine equations involving equal sums of like powers.” American Mathematical Monthly, 75 (10), pp. 1061& (1968).
 Zajta AJ., “Solutions of the Diophantine equation A4 + B4 = C4 + D4.” Mathematics of Computation, 41 (164), pp. 635659 (1983).
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