# Please explain a Mersenne prime **URL:** \ **Category:** Factual Questions\ **Created:** [December 5, 2003, 4:42am UTC](https://boards.straightdope.com/t/please-explain-a-mersenne-prime/217098 "2003-12-05T04:42:11Z")\ **Posts on this page:** 11\ **Page:** 1
**Author:** ![istara](https://avatars.discourse-cdn.com/v4/letter/i/e79b87/32.png) [@istara](https://boards.straightdope.com/u/istara)\ **Post date:** [December 5, 2003, 4:42am UTC](https://boards.straightdope.com/t/please-explain-a-mersenne-prime/217098/1 "2003-12-05T04:42:11Z")
Please explain this in VERY simple terms (if possible) to a non-mathematician. I read about them on here: [http://uk.news.yahoo.com/031202/12/efgb3.html](http://uk.news.yahoo.com/031202/12/efgb3.html) which sort of explains but not fully, and I don’t quite understand it: > [@](#): > > \*\*Prime numbers are positive integers that can only be divided by themselves and one. Mersenne primes are an especially rare type of prime that take the form 2 p -1, where p is also a prime number. The new number can be represented as 2 20,996,011 -1. It is only the 40 th Mersenne prime to have ever been found. \*\* Eg: what’s the 2 represent? ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [December 5, 2003, 4:50am UTC](https://boards.straightdope.com/t/please-explain-a-mersenne-prime/217098/2 "2003-12-05T04:50:49Z")
That should read 2[sup]p[/sup] - 1. Beyond that, it’s self explanatory. Not all 2[sup]p[/sup] - 1 are prime, and in fact, most are not, but if it is, then it’s a Mersenne prime. As stated, _p_ must be a prime number, but it need not be a Mersenne prime. As an example, 2[sup]3[/sup] - 1 = 7, which is prime. 7 is therefore a Mersenne prime. ---
**Author:** ![Dijon\_Warlock](https://avatars.discourse-cdn.com/v4/letter/d/a9a28c/32.png) [@Dijon\_Warlock](https://boards.straightdope.com/u/Dijon_Warlock)\ **Post date:** [December 5, 2003, 4:53am UTC](https://boards.straightdope.com/t/please-explain-a-mersenne-prime/217098/3 "2003-12-05T04:53:24Z")
Mutliplied by two. If you double a prime number p, then double it (2p) and subtract one (-1), and the result is prime, then it (the result) is a Mersenne Prime. ---
**Author:** ![Fang](https://avatars.discourse-cdn.com/v4/letter/f/96bed5/32.png) [@Fang](https://boards.straightdope.com/u/Fang)\ **Post date:** [December 5, 2003, 4:53am UTC](https://boards.straightdope.com/t/please-explain-a-mersenne-prime/217098/4 "2003-12-05T04:53:44Z")
Yahoo poorly formatted the article. It should actually be 2^p - 1, i.e. the number two to the power of a primed number, minus 1. 7 is a Mersenne Prime because it is 2^3 - 1. ---
**Author:** ![Fang](https://avatars.discourse-cdn.com/v4/letter/f/96bed5/32.png) [@Fang](https://boards.straightdope.com/u/Fang)\ **Post date:** [December 5, 2003, 4:54am UTC](https://boards.straightdope.com/t/please-explain-a-mersenne-prime/217098/5 "2003-12-05T04:54:53Z")
Damn you, **Q.E.D** , you and your brilliant examples. 😃 ---
**Author:** ![Dijon\_Warlock](https://avatars.discourse-cdn.com/v4/letter/d/a9a28c/32.png) [@Dijon\_Warlock](https://boards.straightdope.com/u/Dijon_Warlock)\ **Post date:** [December 5, 2003, 4:54am UTC](https://boards.straightdope.com/t/please-explain-a-mersenne-prime/217098/6 "2003-12-05T04:54:59Z")
Ach! Show’s what I don’t know. 2p-1…2[sup]p[/sup]-1…brainfart… ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [December 5, 2003, 5:13am UTC](https://boards.straightdope.com/t/please-explain-a-mersenne-prime/217098/7 "2003-12-05T05:13:52Z")
There’s also a connection between Mersenne primes and even perfect numbers. A _perfect number_ is a number such that, when you sum the divisors of the number, you get twice the original number. 6 is such a number, since the divisors sum to 1 + 2 + 3 + 6 = 12 (twice six). An even number is perfect if and only if it is of the form 2[sup]p-1/sup, where 2[sup]p[/sup]-1 is a Mersenne prime. For example, the first four Mersenne primes are 3, 7, 31, and 127. And so the first four (even) perfect numbers are 2_3 = 6, 4_7 = 28, 16_31 = 496, and 64_128 = 8128. Nobody knows if there are an infinite number of Mersenne primes (or, equivalently, if there are an infinite number of even perfect numbers). Nobody knows if there are any odd perfect numbers. ---
**Author:** ![Heller\_Highwater](https://avatars.discourse-cdn.com/v4/letter/h/278dde/32.png) [@Heller\_Highwater](https://boards.straightdope.com/u/Heller_Highwater)\ **Post date:** [December 5, 2003, 8:38am UTC](https://boards.straightdope.com/t/please-explain-a-mersenne-prime/217098/8 "2003-12-05T08:38:10Z")
Since it was an MSU student whose computer discovered the number, there was an article [here](http://www.statenews.com/article.phtml?pk=21238) State News (Michigan State University’s Student Newspaper) about the discovery. There’s some nice little tidbits about the number itself in the sidebar. One interesting quote from the article was the rate at which the program being used is discovering new primes: > [@](#): > > Kurowski said the GIMPS program has made about six important discoveries on prime numbers. A new record breaks at a rate of about every two years. ---
**Author:** ![tsarina](https://avatars.discourse-cdn.com/v4/letter/t/f4b2a3/32.png) [@tsarina](https://boards.straightdope.com/u/tsarina)\ **Post date:** [December 5, 2003, 3:00pm UTC](https://boards.straightdope.com/t/please-explain-a-mersenne-prime/217098/9 "2003-12-05T15:00:55Z")
**Heller Highwater** , I was just about to post that! Makes you proud to be a Spartan, even though the discovery doesn’t have any practical value. 🙂 ---
**Author:** ![istara](https://avatars.discourse-cdn.com/v4/letter/i/e79b87/32.png) [@istara](https://boards.straightdope.com/u/istara)\ **Post date:** [December 7, 2003, 6:35am UTC](https://boards.straightdope.com/t/please-explain-a-mersenne-prime/217098/10 "2003-12-07T06:35:30Z")
Thanks guys. I actually managed to follow pretty much all of that! BTW **Cabbage** - what is the “point” of a perfect number? Does it have some special usefulness in mathematics? ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [December 7, 2003, 7:14am UTC](https://boards.straightdope.com/t/please-explain-a-mersenne-prime/217098/11 "2003-12-07T07:14:31Z")
> [@](#): > > BTW Cabbage - what is the “point” of a perfect number? Does it have some special usefulness in mathematics? There’s no real point (or special usefulness) to my knowledge. As far as I know, the history of perfect numbers is rooted deeply in numerology. Certain numbers were associated with certain “mystical” properties, and a perfect number was an interesting number, indeed. Perfect numbers have since been considered an interesting topic by mathematicians, but to my knowledge it doesn’t run any deeper than an ancient, interesting curiosity, really. An interesting aside (and I’ll give the caveat that I’m no Biblical scholar, so I may be wrong, I’m just going by what I’ve read other places) is Genesis 32:14: > [@](#): > > 13: And he lodged there that same night; and took of that which came to his hand a present for Esau his brother; **14: Two hundred she goats, and twenty he goats, two hundred ewes, and twenty rams,** 15: Thirty milch camels with their colts, forty kine, and ten bulls, twenty she asses, and ten foals. 220 goats and 220 sheep were offered as gifts. There are (according to what I’ve read) many instances of numerology in the Bible, and this is apparently one of them. A perfect number, remember, is one whose divisors sum to twice the original number. A similar notion is the idea of _friendly_ (or _amicable_) numbers, also dating back to numerology. Two numbers x and y are said to be _friendly_ if the sum of the divisors of x, and the sum of the divisors of y, both sum to x + y. 220 and 284 are such a pair: 220: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 + 220 = 504 284: 1 + 2 + 4 + 71 + 142 + 284 = 504 and 504 = 220 + 284. And so the fact that **220** sheep and **220** goats were offered as gifts is to be taken as a symbol of friendship. One final Biblical reference–note that the creation took place over **six** days (and the seventh was a day of rest). Six–the first perfect number.