Resolutio formulae diophanteae ab(maa+nbb) = cd(mcc+ndd) per numeros rationales
In a previous paper, Euler solved A4 + B4 = C4 + D4 by reducing that problem to solving ab(aa + bb) = cd(cc + dd) in integers. 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 Euler makes the conjecture that two cubes can't sum to a cube, three fourth powers can't sum to a 4th power, and, in general, the sum of n–1 nth powers can't be an nth power. (This conjecture has since been shown to be false.)
Original Source Citation
Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 13, pp. 45-63.
Opera Omnia Citation
Series 1, Volume 4, pp.329-351.