Source: Diophantos, as quoted in The Penguin Book of Curious and Interesting Puzzles, David Wells, 1992, #28.
For a,b,c unique positive integers, and a+b+c < 1000
41, 80, 320 57, 112, 672 88, 168, 273The last question above does not have a simple solution. It is actually just a rewording of an as-yet-unsolved problem (I didn't realize this was the same quesiton when I wrote it.) Thanks to Philippe Fondanaiche for the following discussion:
W.Sierpinski in Elementary theory of numbers (editor:A.Schinzel North-Holland
Mathematical Library):
"We don't know whether there exist three natural numbers a,b,c such the each
of the numbers
a^2 + b^2, a^2 + c^2, b^2 + c^2 and a^2 + b^2 +c^2 is the square of a
natural number.
In other words,we do not know whether there exists a rectangular
parallelepiped whose sides,face diagonals and inner diagonal are all natutal
numbers.
At the opposite, there exist four natural numbers x,y,z,t such that the sum of
the squares of any three of them is a square.
Euler's solution x = 168 y = 280 z = 105 t = 60
Tebay's solution x = 1995 y = 6384 z = 1520 t = 840 "
Let a+b+c = (x+1)^2 Let a+b = x^2 --> c = 2x+1 Let a+c = (x-1)^2 --> b = 4x, a = x^2 - 4x Then we need merely find c+b = 6x+1 to be a square and we find our answer. For x=20, 6x+1 = 121, and c=41, b=80, a=320. For x=28, 6x+1 = 169, and c=57, b=112, a=672. The third answer above is found by letting a+b+c=(x+2)^2 and a+c=(x-2)^2.
For a,b,c unique positive integers, and a+b+c < 1000 41, 80, 320 57, 112, 672 88, 168, 273 The full list of solutions for a,b,c postive integers < 1000: 17, 32, 32 Not unique 41, 80, 320 57, 112, 672 65, 464, 560 >1000 68, 128, 128 Not unique 72, 72, 217 Not unique 88, 168, 273 128, 128, 833 >1000, Not unique 136, 264, 825 >1000 144, 585, 640 >1000 153, 288, 288 Not unique 200, 200, 329 Not unique 272, 512, 512 >1000, Not unique 280, 345, 744 >1000 288, 288, 868 >1000, Not unique 385, 456, 840 >1000 425, 800, 800 >1000, Not unique 720, 801, 880 >1000