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 heard 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 two-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 three-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.
Original Source Citation
Mémoires de l'académie des sciences de St.-Petersbourg, Volume 9, pp. 3-13.
Opera Omnia Citation
Series 1, Volume 5, pp.61-70.