The largest known prime number, discovered by a computer programmer in 2018, is equal to 282,589,933-1 and has 24,862,048 digits. Also known as M82,589,933, it is one of 51 known Mersenne primes. Named for the French priest Marin Mersenne (1588-1648), these primes of the form 2n-1 have dominated prime searches since antiquity. They currently comprise eight of the ten largest known primes, and since the 16th century, the largest known prime has not been Mersenne for a total of only thirteen years.
Early mathematicians studied Mersenne primes for their connection to perfect numbers. Euclid’s Elements tersely states, “A perfect number is that which is equal to its own parts.” In more modern language, it is equal to the sum of its divisors aside from itself. For instance, the divisors of 6 are 1, 2, 3, and 6; as 1+2+3=6, 6 is perfect. These numbers seemed very significant; Augustine (354-430) was not atypical in connecting the six days of creation to the number’s perfection.
In book 7 of his Elements, Euclid shows that if 2n-1 is prime, 2n-1(2n-1) is perfect. His own proof is characteristically geometric, but a simple proof is as follows:
All divisors are either powers of two, or powers of two multiplied by (2n-1). The sum of the first type is (1+2+…+2n-1)=2n-1; the sum of the second type is (1(2n -1)+2(2n-1)+…+2n-2(2n-1))=(2n-1)(1+2+…+2n-2)=(2n-1-1)(2n-1), because 2n-1(2n-1) itself is not counted as a divisor. These two sums add to 2n-1(2n-1), which is indeed the number itself.
Mersenne primes are thus a path to more of those “perfect,” intriguing numbers. But centuries of misconception impaired systematic search. Many thought, for instance, that if n was prime, so was 2n-1 (Mersenne numbers can be easily factored whenever n is composite). And despite the scarcity of known perfect numbers, an impression lasted for centuries that their final digits alternated between 6 and 8. Pietro Cataldi (1548-1626), who proved this last to be false, tried to put together a list of Mersenne primes, but it was only verified up to 219-1. Mersenne himself lent his name to such primes with a list of his own, but it, too, was inaccurate. He knew much of his list was merely speculative—quick techniques for showing primality simply didn’t exist.
Leonhard Euler (1707-1783) made unprecedented progress in this field, as he did in so many others. He showed that all factors of a composite Mersenne number 2n-1 must be one more than a multiple of n, as well as one more or one less than a multiple of 8. Now able to test for divisors more quickly, he found the first new Mersenne prime in almost two centuries: 2,147,483,647, equal to 231-1. He also proved that all even perfect numbers must be of the form 2n-1(2n-1) (no odd perfect number has yet been found, though they may exist). This extended Euclid’s result to get the Euclid-Euler theorem.
By the next century, the fascination with perfect numbers had largely dissipated. More numbers from Cataldi’s and Mersenne’s lists were verified—some as composite, others as prime. Édouard Lucas (1842-1891) found the largest yet—the 39-digit M127—and proposed a general test for Mersenne primes:
Define a sequence S(n) such that S(1)=4 and S(k+1)=(S(k))2-2 for all k. 2n-1 is prime if and only if it is a divisor of S(n-1).
D.H. Lehmer (1905-1991) would finally prove this in the 1930s.
The Lucas-Lehmer test is remarkably straightforward, and proved quite suitable as a program once computers became powerful enough. The first to implement it was Raphael Robinson (1911-1995), a Berkeley professor who tested 2n-1 for all prime n less than 2304. In the year 1952, he found five new Mersenne primes. Many other programmers would find new largest primes in the ensuing decades—even a pair of high schoolers who barely understood the math involved. 38 new Mersenne primes in all were found in the 20th century, as opposed to 2 in the 19th.
Since 1996, GIMPS (Great Internet Mersenne Prime Search) has found all new Mn. Its volunteers use a standard program, which rules out various sorts of composite numbers before applying the decisive Lucas-Lehmer test. GIMPS has continuously been the finder of the largest known prime of any sort; Mersenne primes are, despite their relative rarity, by far the easiest to verify.
The rate of new Mersenne primes has, however, slowed in recent years. 282,589,933-1 has held its record for almost five years, the longest space of time since the 1970s. No one truly knows if the next Mersenne prime is reachable with current computing power. It may not even exist. Despite attempts to graph and predict, our fundamental understanding of Mersenne primes is not substantially different from that of the Renaissance.
Image Credit:
https://commons.wikimedia.org/wiki/File:Marin_mersenne.jpg
https://mersenneforum.org/showpost.php?p=544483&postcount=21
https://t5k.org/notes/faq/NextMersenne.html
Sources:
Dickson, Leonard E. History of the theory of numbers. G.E. Stechert & Co., 1919.
Euclid, Richard Fitzpatrick, and J. L. Heiberg. Euclid’s elements in greek: From Euclidis Elementa. Austin, TX: Richard Fitzpatrick, 2005.
Wow. Cool. Great job!
I’d like to publicly acknowledge the significant time my brother Hudson devoted to make these images as little screwed up as possible.