Squares and Cubes

Considering that 1 is both a square number and a cube, let x be a positive integer such that $a^2 + 1^2 = x = b^3 + 1^3 \iff a^2 = b^3$ . This means for any number that is both a square and a cube, that number plus one satisfies the question although it may not be the least.

A number which is both a square and a cube would be an integer with a power which would be a multiple of 2 (to make it a square) and a multiple of 3 (to make it a cube). The smallest number that satisfies this would be an integer to the power 6. Therefore $a^2 = b^3 \implies a^2 = b^3 = n^6$ . $n$ cannot be 1 because then x would be 2 which the question does not allow so let $n = 2$ . This means that $a = 8$ and $b = 4$ so $8^2 + 1^2 = 4^3 + 1^3 = 65$ . A check of all the numbers below 65 will not give answers to the question so therefore 65 is the answer.