- 2^p - 1
The Reference Frame is the only blog in the world that also informs you that the winner is Curtis Cooper and his new greatest exponent is p = 30,402,457. (Steven Boone became a co-discoverer; note added on Saturday.) You can try to search for this number on the whole internet and you won't find anything; nevertheless, on Saturday, it will be announced as the official new greatest prime integer after the verification process is finished around 1 am Eastern time. If you believe in your humble correspondent's miraculous intuition, you may want to make bets against your friends. ;-)
Actually I am so incredibly sure that you should bet thousands of dollars if someone is ready (and has the courage) to argue that I am mistaken and the exponent won't be 30,402,457. Trust me. Note that we predicted the previous Mersenne prime correctly, too. The new greatest prime looks like follows (more than 9 million digits are omitted):
- 315416475 … 652943871
The exponent may count the number of curves of a given genus in a particular elliptically fibered Calabi-Yau manifold. Or something else.