On the Distribution of Mersenne Divisors By Daniel Shanks and Sidney Kravitz
The Mersenne numbers are those of the form M_{p} =
2^{p} - 1 with p
prime. The only possible divisors of M_{p}
are those of the form 2kp + 1. Let
f_{k}(x) be the number
of M_{p} with
p < x
that have a prime divisor d = 2kp + 1.
As is known, it has not been proven, even for a single k, that
as x . It is also known that
for all k of the form 4m + 2, but, with these values
of k excluded, one expects, heuristically, that (1) is true
for all other k = 1, 3, 4, 5, 7, 8, .... We conjecture, in
fact, a stronger result that includes both (1) for these allowed
k, and (2) for those excluded:
In (3) the product is taken over all odd prime q, if any, that
divide k, and (x) is the
well-known conjectured estimate for the number of twin-prime pairs
< x:
f_{k}(10^{5})
for k < 200. In Table 1 we present a
table of f_{k}(x) for
k
4m + 2 < 60,
x =
10^{5}(10^{5})10^{6}.
The larger range of
Z(x)
could be readily adapted to prove
also, and we prefer to emphasize this relationship.
f_{k}(x)
In Table 2 we list the ratios:
The counts Z(x) were taken from [2],
and are repeated here in Table 3 for convenience.
Table 2 suggests that our conjectures (3) are true
for all
r_{k}(x)
The heuristic argument for (3) is quite convincing,
especially in view of previous successes for similiar arguments. A
Hardy-Littlewood conjecture is
and, similarly, cf. [3], the number of integers n < x such that n and
2kn + 1 are both prime should be asymptotic to
Z(x)
Now the factor
for k = 1, 2, 3, or 4 (mod 4), respectively, and therefore
represents the fraction of the primes 2kn + 1 which have 2 as
a quadratic residue:
Finally, for such a possible prime divisor 2 kn + 1,
we assume that 1/k is the probability that 2 is a (2k)ic
residue of 2kn + 1, for if g is a primitive
root of 2kn + 1, by (9) we have
for some s, and we assume, that the probability of
2k | 2s is 1/k. For these primes, n
and 2kn + 1, we therefore have
2kn + 1 | 2^{n} - 1.Combination of (7), (8), and (4) now yields (3). Now we wish to suggest two extensions of this work to others, since we think these to be of some importance, but are not satisfied with any efforts that we ourselves have made.
(A) We note, first, that only the case
p = 6n + 1 =
a^{2} + 3b^{2},
but only those It is clear, then, that we wish a generalization of the Bateman-Horn conjecture [3], and also its extension by Schinzel [4], to include not only primes but also prime ideals. But we have not satisfied ourselves that we have obtained this with full generality and proper exactitude.
(B) For no It seems to us that this weaker conjecture is provable, but we have not proved it. While (6) has not been proven, one can also examine the sequences
p, p + 2k
David Taylor Model Basin
1. S. KRAVITZ,
"Distribution of Mersenne divisors," Received August 25, 1966. |

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