Question 691044
I am not sure if the problem is meant as a system of non-linear equations, or a simpler problem, looking for integers that satisfy each (or both) equations.
 
The most popular of Pythagorean triples (and the only one I remember) is
(3,4,5) with {{{3^2+4^2=5^2}}} or {{{3^2+4^2=25}}}.
So the only answer to {{{a^2+b^2=25}}} with whole numbers is {{{highlight(3^2+4^2=5^2)}}}. If we allow negative numbers, there are other integer solutions to just that equation (we just cahnge the signs for a and/or b).
 
Curiously, the same numbers 3 and 4 are a solution to {{{a^3+b^3=91}}},
{{{3^3+4^3=27+64=91}}}.
So you could say that the pair (3,4) is the only integer solution to the system
{{{system(a^2+b^2=25,a^3+b^3=91)}}}
 
If we wanted all real numbers that are solutions to the system, we would have to look for two more points where the graphs intersect for
{{{x^2+y^2=25}}} (circle with radius 5) and
{{{x^3+y^3=91}}} <--> {{{y=root(3,91-x^3)}}} (a snakey downwards line).