Primes and perfect square on numbers 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?
Please...

For the second question the answer is no.
The number you want is $222\cdots 111\cdots 000\cdots$ which is divisible by $3$ (because the sum of its digits is $2\cdot 100+1\cdot 100+0\cdot 100=300$ divisible by $3$)
But,it is not divisible by $9$ (because also the sum of its digits is not)
There for it must not be a square.
