I have two equations, which may play an important role in my further studies on theory of numbers.
1) How many pairs of (A, B, x) we can make in $A^x + B ^x = prime$? Here $x$ is $> 2$ and A, B are positive integers.
2) can we find a number(s) with one hundred 0′s, one hundred 1′s and one hundred 2′s be a perfect square. If yes, what is that number or otherwise how to disprove it about such number does not existence?