Matthew Weinstein
Math 301
Abstract:
Mathematicians throughout the ages have studied primes of the form 2p-1, or Mersenne primes, to
identify their distribution and search for new prime numbers. This article details the efforts to identify
this distribution and focuses on three conjectures stated in Wagstaff’s paper, and how these were
discovered. The distributed software search for Mersenne primes is discussed, and how the discovery of
new Mersenne primes advances knowledge of their distribution.
Finding the Next Mersenne Prime
Throughout the ages, mathematicians have been fascinated with prime numbers and
discovering the existence of new primes. Mersenne primes are primes of the form 2p-1. From as early as
the 17th century, mathematicians have been using this equation to identify new prime numbers. With
the advent of powerful computers it is possible to identify prime numbers of enormous size. However,
there is no way to know exactly what will be the next Mersenne prime. Prime numbers do not appear at
regular and predictable intervals. Finding the newest Mersenne primes is an serious challenge, one that
goes hand in hand with the identification of a distribution for these prime numbers. As new Mersenne
primes are found, knowledge of their distribution is advanced.
In the 17th century,the French monk Marin Mersenne took a strong interest in prime numbers.
He searched for a formula to identify all of the primes. Eventually, he realized that prime numbers could
be identified from the formula 2p-1, or calculating various powers of 2 and subtracting 1. Mersenne
identified eleven prime numbers, yet he was mistaken about five of these. In The Music of the Primes,
Sautoy mentions these were the product of “blind guesswork.” (Sautoy, 205) But others give the monk
the benefit of the doubt and suggest the mistakes could be the result of a printing error or attributed to
mathematician, Frenicle De Bessy. However, Mersenne identified quite a few new prime numbers
correctly. However, he did not have the benefit of computers, which now make finding large primes
possible.
With the advent of computers, and a powerful method of testing primality known as the Lucas
Lehmer test, the search for Mersenne primes was underway. In the 1990s, the fastest supercomputers
available, designed by Cray Research, were harnessed to discover new Mersenne primes. In recent years,
however, personal computer users have taken the lead when George Woltman created a program to
test for Mersenne primes and made it available on the internet. It is a distributed programming
project, which means that the problem is broken up into smaller parts and distributed amongst
many participants. This project is known as the Great Internet Mersenne Primes Search, or
GIMPS.
Woltman runs a community-oriented website site for GIMPS, and attracts a huge number of
participants. “Woltman has successfully enrolled over two-hundred thousand computers from all
around the world to create what is in effect one huge parallel-processing machine.” (Sautoy, 208) The
top producers for last year are displayed on a page on the GIMPS website. This acknowledgement of
valuable members of the community and the continued active participation of the founder, George
Woltman, has been very helpful in attracting new contributors and retaining loyal participants who have
been testing primes for many years.
The use of supercomputers, such those built by Cray research, and GIMPS to calculate Mersenne
primes is a method that relies on enormous amounts of calculation, rather than knowledge of the
Mersenne primes distribution, to find new Mersenne primes.
However, the discovery of each Mersenne prime gives us a better picture of their distribution
and this in itself, is an important byproduct of GIMPS. To this day, forty-seven Mersenne primes have
been discovered.
In 1964 D.B. Gillies made a conjecture on the distribution of Mersenne primes. He proposed that
the slope of the distribution of Mersenne primes with respect to the nth prime in the sequence was
linear. The conjecture consisted of three parts of which the first two dealt with the expected number of
Mersenne primes less than a value, x while the second part calculated the number of Mersenne Primes
p
between x and 2x. The third part of the conjecture calculated the probability that 2 -1 is prime.
Gillies found that the probability that a Mersenne number is prime is 2log2 p / p log2. (Wagstaff,
285)
When viewing the graph of the distribution of known Mersenne primes, one will immediately
notice a pattern; it appears linear. The slope approximates the logarithm base two of the logarithm base
two of the nth Mersenne.
From this data, many mathematicians including Wolfgang Erhardt, deduced that the
ratio of two exponents of successive Mersenne primes is 3/2. (Caldwell, Prime Pages) His
results were found using empirical evidence and heuristic data, which was based on the known
values for Mersenne primes.
Later, mathematician Henrik Lenstra realized that Gillies’ conjectures violated the prime
number theorem. In 1980, both Lenstra and Carl Pomerance, working independently, came to
the conclusion that the probability that a Mersenne number is prime is (egamma log ap )/(p log 2)
where a=2 if p=3 (mod 4) and a=6 if p=1 (mod 4). (Wagstaff, 388) This work expanded on and refined
the results of Gillies. Their conclusion was supported with additional empirical evidence and heuristic
data. The ratio of exponents of successive Mersenne primes was determined to be 2 raised to 1/egamma.
It was close 3/2, so again, it agreed very closely with Erhardt’s work.
The distribution of the log of the log of the Mersennes is a Poisson Process. A Poisson Process is
a probability of a number of events occurring in a fixed interval. Because the events occur independently
of one another, the Mersenne numbers could arise at intervals disjoint from one another. The only
factor which determines the probability of the occurrence of a Mersenne prime is the length of the
interval.
When viewing the graph of log2(log2(Mn)) versus n, we can see it is in fact a Poisson Process. The
list of the nth gaps are uncorrelated. The mean and standard deviation also fulfill the requirements for a
Poisson process as they are close in value to the equation determining their distribution, 1/egamma.
(Wagstaff, 385) Dividing the log2(log2(Mn)) scale into twenty-five intervals, we find that the number of
Mersenne primes discovered is close to the number that is expected, with an error of +/-1.
This gives us an indication of where the next Mersenne prime may appear. The probability is
that it will be near 1.47576p, where p is the variable from the formula for Mersenne primes.
Instead of taking into account these theorems about the distribution of Mersenne Primes,
GIMPS does not attempt to pinpoint the location of Mersenne Primes. It takes advantage of the
processing power of thousands of computer-users. The software runs as a background process on a
computer, but the search will still take many weeks or months to be completed. The distributed
software makes use of a computer’s idle cycle, or the computing power that is generally
unutilized by active processes. But that does not mean the search is quick or unobstrustive as
one eager mersenne prime seeker found, in a tale from Music of the Primes. A US telephone
company employee uploaded the program to over 2500 computers at his company. The FBI
tracked the source of the hold-up when the branch in Phoenix was taking five minutes to
retrieve phone numbers instead of five seconds, and the employee was summarily fired.
GIMPS administrators take into account the processing power of computers in
delegating searches to participants. Low end P-3’s and Athalons are assigned the task of trial
factoring. In this portion of the search, the software uses small divisors to eliminate non-prime
candidates. Faster machines are given the far more taxing task of performing the Lucas Lehmer
test. The Lucas Lehmer test checks primality by eliminating large factors. According to the
GIMPS help page, “1.4GHz and faster computers get first-time primality tests” (Great Internet
Mersenne Prime Search).
The difficulty with finding a distribution for prime numbers stems from their unpredictable
nature. However mathematicians have experienced success in estimating where the next prime will
occur, in a similar fashion to the way that the distribution of Mersenne Primes was found using empirical
evidence and heuristic methods. For large prime numbers, Gauss determined that the number of primes
in the neighborhood of x could be found with 1/ln x. French mathematician, Legendre determined that
Pi(x), the number of prime numbers less than x, could be approximated by x divided by ln x – B, where B
is some constant. Legendre and Gauss’ results are equivalent and are referred to as the prime number
theorem. This result was used by Gillies, Pomerance, and Lenstra in their conjectures on Mersenne
prime distribution.
For the Mersenne Prime, 2p-1, to be prime, p must be prime as well. The conjectures in
Wagstaff’s article show that the probability of a Mersenne prime occurring can be shown by applying
the prime number theorem. When this result is used to manipulate 2p-1, the result is 1/log(2p-1). This
would be the estimation used for the distribution, but it disregards an important aspect of Mersenne
Primes: they do not behave like random numbers. In fact, they can only be divisible by primes of the
form 2mp+1.
The formula must be multiplied by 2, because 2p-1 cannot be even. And it cannot be divisible by
3, so it must be multiplied by 3/2. If q is the number of prime number is less than the Mersenne number,
the equation must be multiplied by q/(q-1) = 1/(1-1/q), arriving at this result. (Wagstaff, 386)
1/(k log 2) . 2/1 . 3/2 . 5/4 . ... . q/(q-1)
This is estimated as egamma . log(2k)/(k log 2), using Merten’s theorem. (Ribenhoim, 95) The
probability that p is prime and the probability that 2p-1 is prime are then combined into one equation,
resulting in egamma/(k log 2). This conclusion was used to derive the first of the three conjectures in
Wagstaff’s paper. With this explanation, to gain clarification about how the calculation of Mersenne
prime distribution was arrived at.
The search for very large Mersenne primes with GIMPS goes hand in hand with mathematicians’
study of their distribution. Each discovery of a new prime provides data which can then be of use in
creating theorems about how Mersenne primes appear. And with knowledge of the distribution, seekers
of new Mersenne primes have access to knowledge which may be of use in finding new primes. Perhaps
as new primes are found, a more distinct pattern will be discovered, and the Mersenne Prime
distribution will be known with the greatest accuracy possible.
References
[1] David Wells. Prime Numbers: The Most Mysterious Figures in Math. 1st Edition: Wiley,
2005.
[2] Marcus du Sautoy. The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics.
HarperCollins, 2003.
[3] Gunter M. Ziegler. “The Great Prime Number Record Races" Notices of the AMS April,
2004 Vol. 51 No. 4.
[4] P. Ribenhoim. The New Book of Prime Number Records. 3rd Edition: Springer-Verlag, 1996
[5] Samuel S. Wagstaff Jr. “Divisors of Mersenne Numbers" Mathematics of Computation January, 1983 Vol. 40 No. 161.
[6] Chris Caldwell. “Mersenne Primes: History, Theorems, and Lists." URL: http://primes.utm.edu/mersenne/.
[7] “Great Internet Mersenne Prime Search." February 23, 2011. URL: http://www.mersenne.org/default.php