[thomasahle]

One application is that it's fairly cheap to take `n mod p` when `p` is on the form $2^q-1$. In particular

   x ≡ x mod 2^q + ⌊x/2^q⌋ (mod p).

If you are making a hash function, and need to take mod a prime, this is useful.

Another application of slightly larger Mersennes is https://en.wikipedia.org/wiki/Mersenne_Twister

[kristianp]

I imagine these record-breaking primes are too large to be useful in a hash function though.

[thomasahle]

Certainly. These are just for fun, or maybe in the future giving some motivation for some number theory conjectures. Who knows.