E763  De tribus pluribusve numeris inveniendis, quorum summa sit quadratum, quadratorum vero summa biquadratum
(On finding three or more numbers, the sum of which is a square, while the sum of the squares is a fourth power)
Summary:
Euler attributes this problem to Fermat and says he got it from Lagrange, to find three numbers, x, y, and z such that x + y + z is a square and xx + yy + zz is a fourth power. He starts with a 2variable problem, asking that x + y be square and xx + yy a fourth power. He finds a pair of numbers in the trillions. Then he finds a 3variable solution, x = 49, y = 64, z = 8 and goes on to four variables, x = 193, y = 104, z = 48, v = 16 and even five variables.
Publication:

Originally published in Memoires de l'academie des sciences de St.Petersbourg 9, 1824, pp. 313

Opera Omnia: Series 1, Volume 5, pp. 61  70
 Reprinted in Commentat. arithm. 2, 1849, pp. 397402 [E763a]
Documents Available:
An English translation of E763, by Christopher Goff, is available.
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