#### English Title

Resolutio formulae diophanteae *ab*(*maa*+*nbb*) = *cd*(*mcc*+*ndd*) per numeros rationales

#### Enestrom Number

716

#### Fuss Index

54

#### Original Language

Latin

#### Published as

Journal article

#### Published Date

1802

#### Written Date

1778

#### Content Summary

In a previous paper, Euler solved *A*^{4} + *B*^{4} = *C*^{4} + *D*^{4} 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 *n*th powers can't be an *n*th 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.

#### Record Created

2018-09-25