>>12417820So what's the difference between squares and cubes then? Certainly from Pythagorean triples you know that the sum of two squares can equal a square; e.g. 9 + 16 = 25 or 25 + 144 = 169. Why can't the sum of two power of integers share the same power of another integer only when the power is larger than 2?
You can't just use "intuition" in number theory. For example, try to solve the equation x^3+y^3+z^3 = 33 where x,y,z are integers. You might try a lot of examples and none of them work, so you could conclude that there's probably no solutions. But there is a solution, and it is
x = 8866128975287528
y =-8778405442862239
z = -2736111468807040
This solution was only discovered in March 2019, using very advanced number theoretic techniques.
