## Enestrom Number

702

## Fuss Index

42

## Original Language

Latin

## Content Summary

Euler searches for all integers *N* such that the formulas *A*^{2}+*B*^{2} and *A*^{2}+*NB*^{2} can both be squares at the same time. By putting *A* = *xx*–*yy* and *B* = *2xy*, the first expression becomes a square; to make the other one a square also, one takes the *A*^{2} to be *zz* and obtains (*x*+*y*)/*z* Â± *xx*/(*zz*), and the question reduces to finding values for *z* such that *N* becomes an integer. He finds, among the first 100 natural numbers, the following values for *N* that satisfy the problem: 7, 10, 11, 17, ....

## Published as

Journal article

## Published Date

1798

## Written Date

1777

## Original Source Citation

Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 11, pp. 78-93.

## Opera Omnia Citation

Series 1, Volume 4, pp.255-268.

## Record Created

2018-09-25