Fermat's last theorem is a mathematical statement that is mathematically proven true.

You can also consider the case you mentioned in your example: a = b = 1/4. Then you get a^2 + b^2 = 1/8 = c^2 which gives c = 1/(2 sqrt(2)), which is also not an integer.

Actually any case a = b = p/q != 0 doesn't work, as a^n + b^n= 2 p^n / q^n = c^n implies that c = (2^(1/n)) (p / q). As 2^(1/n) is irrational for every n > 1 and p / q is rational we have c irrational, so also not an integer (if rational != 0 then rational * irrational = irrational).