Assuming that the prime exponent of the next unknown 52^{nd} Mersenne prime (if any) can be represented as a sum of k smaller known prime exponents, the number of prime exponents to be tested is reduced roughly by an order of magnitude as compared to the total number of prime exponents. Eliminating the prime exponents that have already been verified/factored, said number is reduced roughly by two orders of magnitude.
k, Number of Prime Exponents (repetition of same exponents is allowed in the summation of k smaller exponents), Number of Remaining Untested/Unverified Prime Exponents
2, 5, 0
3, 3162, 203
4, 2801, 193
5, 306293, …
6, 303972, …
7, 8674167, …
8, 8685411, …
k, Number of Prime Exponents (repetition of same exponents is not allowed in the summation of k smaller exponents), Number of Remaining Untested/Unverified Prime Exponents
2, 5, 0
3, 2952, 179
4, 2412, …
5, 224493, …
6, 214739, …
7, 5541799, …
8, 5500506, …
Note: The ellipsis indicates that it is preferable not to congest the server in checking the status (Untested or Unverified) of so many prime exponents. One could check specific narrow ranges instead.
