Solution #23

2¹⁰²⁴-1 is a difference of squares, so it can be factored as (2⁵¹²+10(2⁵¹²-1).

2⁵¹²-1 is again a difference of squares, so it can be factored as (2²⁵⁶+1)(2²⁵⁶-1).

Again we have a difference of squares. Continuing in this manner, we eventually get to 2⁴-1, which factors as (2²+1)(2²-1) = 5*3. So 5 and 3 are both prime factors. Since 2¹⁰²⁴-1 is odd, 2 is not a prime factor of it, and 5 and 3 must be the smallest prime factors. Thus the answer is 5*3 = 15.