The Number of Prime Divisors
of Certain Mersenne Numbers*
By John R. Ehrman
It has been conjectured by Gillies [1] that if Mp = 2P — 1 is the Mersenne
number for some prime p, and if A < B ^ V Mp as B/A and Mp —* <=°
f then the
number of prime divisors of Mp in the interval [A, B] is Poisson distributed, with
mean
, s
m ~ log (log B/log A )
m ~ log (log B/log 2p)
if
A ^2p,or
if A < 2p .
It is the purpose of this paper to describe two tests of a modified form of this
conjecture.
It is known that all divisors of Mp must be of the form 2kp + 1 and simultaneously of the form 8k' ± 1, where k and k' are arbitrary integers. Also, the
prime divisors of Mp may be of one of the forms An + 1 or An + 3. Thus if p =
An + 1, the smallest possible divisor q is 6p + 1, and if p = An + 3, the smallest
possible divisor is q = 2p + 1. Thus Eq. (1) is modified slightly: the expected
number of prime divisors of Mp in the interval [Q, B], where Q is not less than the
smallest possible divisor of Mp, Q < B ^ V MP, and as B/Q, Mp —-><x>
¡ is Poisson
distributed with mean
(2)
mQ~ log (log .B/log Q) .
Since the observed results in a group are drawn from two populations corresponding to the two forms of p, there is a question as to what value m should be
used for the estimated mean number of divisors. It would be possible, for example,
to separate the two populations and test the samples independently. It was felt,
however, that a fuller test of the applicability of the conjecture (2) could be made
by testing all primes with no distinction as to form.
In calculating an estimate of the mean m to be used in statistical tests, it was
noted that
(a) the sum of two independent random variables from Poisson distributions
with parameters mi and m2 has a Poisson distribution with parameter (mi 4- m2) ;
(b) if v(x; k, t) is the number of primes p = t (mod k) which do not exceed x,
and if (k, t) = 1, then [2]
v(x; k, t) ~ v(x)/d>(k) .
This means that one may expect nearly equal numbers of primes of the forms
An + 1 and An + 3 in a large sample of primes; this is the justification for not distinguishing the primes as to form.
Thus an unbiased asymptotic estimate of the mean may be taken to be
Received November 9, 1966. Revised February
* Work supported
by U. S. Atomic Energy
13, 1967.
Commission.
700
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PRIME DIVISORS OF MERSENNE
(3)
NUMBERS
701
m = \ (m2p+i + m6p+i) .
Thus, for example, in the interval 100000 < p < 102500, it is found that m2p+i =
0.5645 and mep+i = 0.4784, so that the Mersenne number corresponding to a prime
drawn at random in the interval would be expected to have an average of 0.52
divisors less than 231.
The tests performed on the results given in [3] were a test of the mean number
of divisors, and a test of their Poisson distribution. Because the change in p over
each of the intervals tested is relatively small, the value of p used in computing m
from Eq. (3) was simply the midpoint of the interval in p from which the sample
was drawn.
A program was written for an IBM System/360 (Model 50) computer which
tested for divisors of Mp using the congruence
2P = 1
(mod q) .
The test was coded [4] in the following manner :
1. In binary form, p = JA¡_0ai2i, and 2" = 11%o (22V¡.
2. Let Ri = 22i (mod q) = R\_x (mod q), and Sj = Ip:_0 (A¿)ai (mod q).
Thus Sj need be computed
from Sj-i only if a,- = 1.
3. If Sn = 1, q\Mp.
A. The first five steps of the calculation may be done in one step by taking the
five low-order bits of p to compute Ri = 2p(mod32).
Divisors q < 231were computed for 100000 < p < 300000. To compare the
observed distribution of primes with that predicted by Eqs. (2) and (3), the values
of p were grouped so that p fell into one of the 80 groups defined by
1000004- 2500* < p < 102500 + 2500¿,
i = 0(1)79. In each group, the total number of primes observed and the number of
primes with j divisors were counted. These results are tabulated in Table I. For
each p which has one or more divisors q < 231, the value of p and the associated
values of k = iq — l)/2p are tabulated in [3].
To test the estimate of m, N samples of p were observed between limits L and
U, where N = ir({7) — 7r(L), and L < p < U. The total number of divisors T
was counted, and the sample mean x = T/N was computed. The sample variance
was found from
I
*
2
1
6
s2 = a Z) D2 - (x)2 = •-- Ç n2Kn- (xf ,
where D¡ is the number of divisors observed for the jth prime in the sample, and
Kn is the number of Mersenne numbers in the interval with n divisors. (Because
of the method used, no tests were made for multiple factors.) As the number of observations becomes large, it is expected that the variable
t=
(N -
l)l/2(x
- m)/s
should become normally distributed (0, 1). The observed values of N, D, and t for
each group are given in Table I. The expected number of divisors E is simply the
product of m and N.
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702
J. R. EHRMAN
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703
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704
J. R. EHRMAN
To test the hypothesis of Poisson distribution, a chi-squared test was performed
on the observed distribution of divisors. The counts were put in three classes: no
divisors, one divisor, and two or more divisors; the numbers of primes with i
divisors are listed in the columns K¿ for i = 0(1)5. (See also reference [5].) The
computed values of chi-squared for each of the eighty groups are given in Table I.
The values of chi-squared were computed from the formula
x2 = (Ne~m - Ko)2 + (Nme~m - Ki)2
+ (N(l
- e~m - me~m) - K2 - Ks - K* - K5)2,
and are given in Table I in the column headed x2To test for the possibility that distinguishing between primes of the form
An + 1 and An + 3 might lead to significantly different results, t and x2 were also
computed for m = (l/N^Nim^p+i + (N — Ni)m2p+i], where Ni is the number of
primes p = 1 (mod 4) observed in the interval. The average values of t and x2
obtained were slightly larger than those given at the end of Table I.
A comparison of the expected and observed distributions of t and x2 is given in
Table II. The agreement is seen to be satisfactory.
Table
II
Observed distribution of t and chi-squared.
In both cases, the expected number of values in the ranges indicated is 10.
Upper Limit
on t
-1.15
- .674
- .319
0.0
+ .319
+ .674
4-1.15
Number
of Values
Upper Limit on
Chi-Squared
Number
of Values
5
11
0.266
0.576
0.940
1.386
1.962
2.772
4.158
10
12
9
10
7
10
13
8
12
14
I wish to thank J. C. Butcher for several stimulating
and the referee for several suggestions.
Stanford
Linear Accelerator
14
9
and helpful discussions,
Center
Stanford University
Stanford, California
1. D. B. Gillies,
"Three new Mersenne primes and a statistical
theory,"
Math. Comp., v.
18, 1964, p. 94; Also, Digital Computer Laboratory Report No. 138, Univ. of Illinois, Urbana,
111.MR 28 #2990.
2. W. J. LeVeque, Topics in Number Theory, Vol. 2, Addison-Wesley, Reading, Mass., 1956,
p. 252. MR 18, 283.
3. J. R. Ehrman, "Prime divisors of Mersenne numbers," TN-66-40, Stanford Linear Accelerator Center, Stanford, Calif.
4. Sidney Kravitz,
"Divisors of Mersenne numbers 10,000 < p < 15,000," Math. Comp.,
v. 15, 1961, p. 292. MR 23 #A833.
5. Review No. 113, Math. Comp., v. 19, 1965, p. 686.
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