Whether there exist subexponential-size locally decodable codes, and sub-nε-communication private information retrieval (PIR) protocols, have been major open problems for a decade. A new preprint by Sergey Yekhanin reveals that both of these questions hinge on — wait for this — whether or not there are infinitely many Mersenne primes. By using the fact (discovered a month ago) that 232,582,657-1 is prime, Yekhanin can already give a 3-server PIR protocol with communication complexity O(n1/32,582,658), improving the previous bound of O(n1/5.25). Duuuuuude. If you’ve ever wondered what it is that motivates complexity theorists, roll this one up and smoke it.