 In other words, the number of Mersenne primes with exponent "p" less than "y" is asymptotically

This means that there should on average be about formula_4 ≈ 5.92 primes "p" of a given number of decimal digits such that formula_5 is prime.The conjecture is fairly accurate for the first 40 Mersenne primes, but between 220,000,000 and 285,000,000 there are at least 12, rather than the expected number which is around 3.7.More generally, the number of primes "p" ≤ "y" such that formula_6 is prime (where "a", "b" are coprime integers, "a" > 1, −"a" < "b" < "a", "a" and "b" are not both perfect "r"-th powers for any natural number "r" > 1, and −4"ab" is not a perfect fourth power) is asymptotically

where "m" is the largest nonnegative integer such that "a" and −"b" are both perfect 2"m"-th powers.