2012/024) Prove there are no positive integer solutions to x^2 + x + 1 = y^2?

we know x^2 < x^2 + x + 1 < x^2 + 2x + 1 as x is positive

so x^2 < x^2 + x + 1 < (x+1)^2

as x and x +1 are successive integers there cannot be an integer whose square is between x^2 and (x+1)^2,
hence proved