THREE NEW MERSENNE PRIMES, AND A CONJECTURE,

The Mersenne primes, 9689, 9941, and 11213, are discussed and the following conjecture is formulated If A is less than or equal to B is less than or equal to the square root of Mp where A and B are integers and Mp is a Mersenne prime, then the probability that there is no divisor s of Mp in the interval A,B, is given asymptotically by logALog B if A is greater than or equal to 2p. or by log2plogB if A is less than 2p and prime divisors are statistically inde pendent. Thus Mersenne numbers are asserted to have the same likelihood of being prime, and to have the same statistical distribution of factors as integers of about the same size which are selected to have no factors less that 2p but are otherwise randomly chosen. Author