 89 and 107, which were missed by Mersenne, satisfy the second condition but not the other two.Mersenne may have thought that 2"p" − 1 is prime only if "p" = 2"k" ± 1 or "p" = 4"k" ± 3 for some natural number "k", but if he thought it was "if and only if" he would have included 61.The New Mersenne conjecture can be thought of as an attempt to salvage the centuries-old Mersenne's conjecture, which is false.