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)


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 2-variable 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 3-variable 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.

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