Some number theory fun with the taxicab number

Notice that if $p$ is a prime, and $p-1 | 1728$ , then $n^{1728} \equiv 1 \pmod p$ , implying $n^{1729} \equiv n \pmod p$ , or that $n^{1729} - n \equiv 0 \pmod p$ .

Note, however, that even if $p-1 | 1728$ , then if $p|n$ but $p^2 \nmid n$ , then $n^{1729} - n$ is not divisible by $p^2$ .

This implies $d$ is squarefree.

Thus, the primes which divide $d$ are those for which $p-1 | 1728$ . UtiLiZiNG wOlFraM aLpHA, we can see that the primes are $2,3,5,7,13,17,19,37,73,97,109.193,433,577$ . Notice that $d$ is the product of these, thus since $d$ has $14$ primes in its factorization, $d$ has $2^{14} = \boxed{16384}$ positive divisors.