Five interesting things about Mersenne primes

A Mersenne prime is a prime number that is one less than a power of 2. These primes are indexed by the corresponding power of two, i.e. Mp = 2p − 1. It turns out p must be prime before 2p − 1 can be prime.

Here are five things I find interesting about Mersenne primes.

1. Record size primes

The largest known prime number is a Mersenne prime, M82,589,993, proved prime in 2018. And ever since M521 was proven prime in 1952, the largest known prime has always been a Mersenne prime (with one exception in 1989). See a history of prime number records.

One reason for the prevalence of Mersenne primes in the record books is that there is a special algorithm for testing whether a number of the form 2p − 1 is prime, the Lucas-Lehmer test.

2. Finiteness

There may only be a finite number of Mersenne primes. Only 51 are known so far. There are conjectures that predict there are an infinite number of Mersenne primes, but these have not been settled.

3. Connection with perfect numbers

Euclid proved that if M is a Mersenne prime, M(M + 1)/2 is a perfect number. Two millennia later, Euler proved that if N is an even perfect number, N must be of the form M(M + 1)/2 where M is a Mersenne prime. (More details here.)

Since we only know of 51 Mersenne primes at the moment, and we don’t know of any odd perfect numbers, there are only 51 known perfect numbers.

4. Connection with random number generation

The Mersenne twister is a popular, high-quality random number generator. The generator is so named because its period is a Mersenne prime, M19,937.

5. History

Mersenne primes are named after the French monk Marin Mersenne (1588–1648) who compiled a list of Mersenne primes. Mersenne wasn’t the first to be aware of such primes. As mentioned above, Euclid connected these primes with perfect numbers.

Marin Mersenne is one of my academic ancestors. I studied under Ralph Showalter, who studied under Tsuan Ting, and so forth back to Frans van Shooten Jr., who studied under Marin Mersenne.

What I find fascinating about this is not my particular genealogy, but that adequate records exist to construct such genealogies. The Mathematics Genealogy Project has over 150,000 records, some reaching back to the Middle Ages.


14 thoughts on “Five interesting things about Mersenne primes

  1. I find it remarkable that Martin Mersenne is even in the Mathematics Genaology project, not because of the existence of records, but because he never was awarded, nor conferred a Ph.D. in any field, let alone mathematics. Yet, to be included today one must have a Ph.D. awarded by a university with a Ph.D. holder as an advisor, and a dissertation.

    Presumably Mersenne is included because of his contributions to mathematics, and if so I find it even more remarkable that modern members of the genealogy are not included on the basis of contributions to any field, but solely by the conferral of a Ph.D. degree. Mathematics for Mersenne seems to have been incidental to his main work in acoustics, music theory, philosophy, and theology. If so, this makes his inclusion in the genealogy even more remarkable yet. I’d wager that no one today who works with mathematics as an avocation and who does not have a Ph.D. would even be considered for inclusion in the genealogy.

    The closest thing that Mersenne had to a Ph.D. that I found in the Wikipedia article is that he received his full holy orders in 1613. On the other hand, I think there is more than a passing resemblance between ordination and conferring a Ph.D. today. The Mathematics Genealogy Projects is really a record of apostolic succession.

    On fascinating records that exist, my (human) genealogy traces back to Charlemagne. Charlemagne has a genealogy that traces all the way back to Adam. Yes, that Adam, mentioned in Genesis. So I have a nifty stack of papers which documents my unbroken descent from Adam. How’s that for records whose existence is remarkable?

  2. John: I have heard that a German branch of my family tree also traces back to Charlemagne. So we may be cousins. :) Though as Americans of European descent, I imagine you and I have a common ancestor closer than Adam or Charlemagne.

    As for Mersenne’s credentials, I doubt many of the medieval scholars would have considered themselves mathematicians. Some certainly would not. I wonder when mathematician as a specialization even appeared. Probably 19th century. I imagine many of the folks we call mathematicians would have called themselves natural philosophers. So I wouldn’t interpret the arrows in the graph as meaning “supervised a math PhD” but only that one person supervised or mentored another. Maybe you have a theologian who supervised a philosopher who supervised a scientist etc.

    I agree that the credentialism in the genealogy is somewhat arbitrary. But if you’re going to draw a graph like this, you’ve got to have some objective definition, and apparently doctorial degrees are astonishingly well documented. Influence is perhaps more important, but harder to document. I imagine many for many people, their official advisor was not the person who influenced them most.

  3. John,

    I agree that doctoral degrees are useful because they are well documented. After all, at night you always look under the streetlights for your lost keys, right? ;-) What I find suprising is their use in attempting to document the intellectual history of mathematics. People who study the history of nations, cultures, and so on only rarely use human genealogy to document it. It seems a bit like using records of apostolic succession to document the history of theology, or even the history of the Catholic Church. Sure, in the strict sense it is part of it, but I doubt it sheds much light on anything other than itself. I wonder if anyone has attemped to study how many Ph.D. holders still work in the field of their dissertation five or ten years after consigning it to the lightless basement of the university library. It would be even more interesting to find out how many are even still publishing or participating in the academy at all.

    Maybe I’ve misjudged the Mathematics Genealogy Project. For a while I’ve considered it to be like a parlor game (like one’s Erdos number or the degrees of Kevin Bacon), or a source of factoids, or something along those lines.

    But our recent conversations about the dominance credentials in the academy, and my own speculations about the quasi-religious significance of their conferral, has led me to a new understanding.

    The Mathematics Genealogy Project is not a genealogy of mathematicians, or professionals of any type!

    It is a genealogy of credentials!

    Specifically, it is a genealogy of mathematics related university-granted dissertation supported Ph.D.s.

    So it is actually no surprise whatsoever that it would prove popular among academics, even though it has so little relation to the field of mathematics per se. Academics is nothing without credentials, so why not construct a genealogy of credentials? You can trace which credential begat your credential, and which credential begat that one, and so forth. For a group in which credentials are everything, there could scarcely be a more relevant genealogy.

    Speaking of Charlemagne, I believe there is a society of his descendants.

  4. Dr Shirleen Stibbe

    And here’s a 6th:

    In 1644 Mersenne claimed in his Cogitata Physico Mathematica, that for all primes, the integer 2^p – 1 is prime if and only if p is 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 or 257. He was wrong!

    At a research meeting in 1903, Professor F.N. Cole calculated, by hand, the value of 2^67 – 1 on one half of a blackboard and wrote down the integers 193707721 and 761838257287 on the other. He then, by hand, calculated the product of these two integers to be 2^67 – 1. He sat down to prolonged applause having said nothing. He later revealed that finding two factors of 2^67 – 1 had cost him 3 years of Sundays. (Some references say 20 years of Sunday afternoons)

    Gridgeman, N (1963), “The search for perfect numbers”, The New Scientist (334): 86–88

  5. If anyone reading this is interested in discovering the largest Mersenne Prime numbers *ever*, just google “CEMPLLA”.

    It’s just been released, and is the only software in the world (at the time of this writing) that can (feasibly) discover Mersenne Primes up to and including a *billion* or more decimal digits.

    It’s completely documented and easy to use, but only runs under Windows 64-bit operating systems, and requires one or more (preferably more) NVidia brand GPUs to work. It basically exploits the computing power of every connected NVidia brand GPU in your system to perform the required “Lucas-Lehmer Test” on Mersenne Prime candidates or your choosing.

    So happy computing!

  6. Please update the article. Volunteers working together worldwide have eliminated many composite Mersenne numbers of prime exponent, through trial factoring, P-1 factoring, or primality testing, and the highest known Mersenne prime is now the 50th known, with exponent 77,232,917, found Dec 26 2017. The best available algorithms run on the most efficient software and modern hardware (multicore cpus or massively multicore gpus) scale as n squared times log n times log log n for primality test run time on one exponent’s primality test. (See the published and peer reviewed articles on fast multiplication of very large numbers using fft transforms.) Finding the next is inherently parallel, since many prime exponents must be tested to find one new prime mersenne number.

  7. Hi,

    In the 2nd interesting thing states that 47 Mersenne primes are known so far, while at the 3rd point the number is 51.

    Something else, regarding the conversations in the Comments section with mr John Venier: We aren’t supposed all descendants off Adam..?

  8. Thanks. I had updated part of the post when the latest Mersenne prime was discovered, but didn’t update the count. As of December 2018, we’re up to 51.

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