Now for something completely different: Compute $\binom{2^{32}-1}{2^{31}} \bmod{p}$ where $p = 3\cdot2^{29}+5$,
efficiently. Unfortunately, all that I can come up with uses at least $\operatorname{O}(p)$ multiplications and divisions. The shortcuts by
Granville et al only help for powers of small $p$.