The numbers 184 and 345 have a special property. Their sum, the sum of their squares, and the sum of their cubes are all perfect squares:
184 + 345 = 23^2
184^2 + 345^2 = 391^2
184^3 + 345^3 = 6877^2

Find another primitive pair of non-zero integers with the same property. Note, a primitive solution is a solution which is not a multiple of any smaller solution.
If you have extended precision math software, try to find a third or fourth primitive solution.

Tip: one of the numbers may be negative.

Some solving tips:

Equation 2 (a^2+b^2=y^2) is a Pythagorean equation.  The general parameterization is [a=k*(x^2-y^2), b=k*(2*x*y), y=k*(x^2+y^2)]

Start with that and find a way to easily find k so that Equations 1(a+b=x^2) and 2 is satisfied simultaneously.  Then move on to equation 3 (a^3+b^3=z^2).

Posted by Brian Smith on 2008-04-24 00:42:31