 It is also a conjecture that no number which is greater than 127 satisfies all three conditions.As of February 2020 all the Mersenne primes up to 243112609−1 are known, and for none of these does the third condition hold except for the ones just mentioned.Primes which satisfy at least one condition are

Note that the two primes for which the original Mersenne conjecture is false (67 and 257) satisfy the first condition of the new conjecture (67=26+3, 257=28+1), but not the other two.