Fermat's Last Theorem

According to Fermat's Last Theorem, no three positive integers a, b, c satisfy the equation, for any integer value of n greater than 2. For n = 1 and n = 2, the equation have infinitely many solutions.
 

Some solutions for n = 1 are,
 2 + 3 = 5
 7 + 13 = 20
 5 + 6 = 11
 10 + 9 = 19

Some solutions for n = 2 are,

No counter example within given range and data

Time Complexity: O(m²logn) , where m is the limit
Auxiliary Space: O(1)

Please suggest if someone has a better solution which is more efficient in terms of space and time.