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Biographies of Mathematicians-Marin Mersenne

(1588-1648)

His Life

Marin Mersenne was a 17th century monk and mathematician, who mainly studied the numbers 2p - 1.

Marin Mersenne is best known for his role as a sort of clearing house for correspondence between eminent philosophers and scientists, and for his work in number theory.

He was born on September 8, 1588, in Maine, France. Not much is known about his younger life at home. However, since his father was a laborer, it seems as if Mersenne lived a relatively poor life as a young boy. Mersenne attended his grammar studies at the College of Mans in Paris, France. How this schooling was financed is not known. However, from 1604 to 1609 he was instructed at the Jesuit College at La Fleche, France. The Jesuits probably financed his education there. Then, from 1609 to 1611, he studied theology at Sorbonne University in France. It would seem right to say that Mersenne got at least the equivalent to a B.A. In 1611, Mersenne joined the religious order of the Minims. The Minims thought that they were as the least (minimi) of all the religions on earth, and they devoted themselves to prayer, study, and scholarship. Mersenne continued his education at Nigeon, France, and then at Meaux, France. After his return to Paris in 1612, Mersenne became a priest at the Place Royale. From 1614 to 1618, Mersenne taught philosophy in Nevers, France in a Minim convent. In 1619 he went back to Paris to the Minims de l'Annociade near Place Royal. Mersenne had many meetings in his home with Fermat, Pascal, Gassendi, Roberval, Beaugrand, and others who became prominent mathematicians such as Mersenne. He also talked and interacted with other well-known mathematicians. We will talk about his accomplishments in his later years in the next section.

Mersenne's Accomplishments

Many early writers felt that the numbers of the form 2p - 1 were prime for all primes p, but in 1536 Hudalricus Regius showed that 211 - 1 = 2047 was not prime. By 1603 Pietro Cataldi had correctly verified that 217 - 1 and 219 - 1 were both prime, but then incorrectly stated 2n-1 was also prime for 23, 29, 31 and 37. In 1640 Fermat showed Cataldi was wrong about 23 and 37; then Euler in 1738 showed Cataldi was also wrong about 29. Sometime later Euler showed Cataldi's assertion about 31 was correct.

Marin Mersenne investigated prime numbers and he tried to find a formula that would represent all prime numbers.

Although he failed in this, his work on the numbers 2p - 1, p prime has been in continuing interest in the investigation of large primes. It is easy to prove that if the number n = 2p - 1 then p must be a prime. In 1644, Mersenne stated in the preface to his Cogitata Physica-Mathematica (1644) that the numbers 2n - 1 were prime for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 were composite for all other positive integers n < 257.

Mersenne numbers are of the form 2p - 1. Mersenne Primes are primes of the form 2p - 1.

Over the years, it has been found that Mersenne was wrong about five of the primes less than or equal to 257. He claimed two that didn't lead to a prime, 67 and 257, and missed three that did, 61, 89, and 107. In 1633, Mersenne published Traite de mouvments, and in 1634 he published Les Mechanique de Galile, which was a version of Galileo's lectures on mechanics. He translated parts of Galileo's Dialogo into French and in 1639 he published a translation of Galileo's Discorsi. It is through Mersenne that Galileo's work became known outside of Italy. Two important publications in mathematical physics by Mersenne were L'Harmonie Universelle in 1636, and, as mentioned above, Cogitata Physico-Mathematica in 1644. Mersenne also wrote Traite d'Harmonie Universelle in 1627, a work on music, musical instruments, and acoustics. After his death, letters from people such as Fermat, Huygens, Pell, Galileo, and Toricelli were found in his house.

Applications and Uses of Mersenne's Accomplishments

Over the years, Mersenne's accomplishments have been the subject of many researches. It was obvious to Mersenne's peers that he could not have tested all of these numbers (in fact he admitted as much), but they could not test them either. It was not until over 100 years later, in 1750, that Euler verified the next number on Mersenne's and Regius' lists, 231 - 1, was prime. Euclid showed that the known perfect numbers were of the form 2p-1*(2p - 1). Euler proved that even perfect numbers could only be in this form. It remains an open question as to whether there are any odd perfect numbers. Whenever another mersenne prime is found, another perfect number is generated. After another century, in 1876, Lucas verified 2127 - 1 was also prime. Seven years later Pervouchine showed 261 - 1 was prime, so Mersenne had missed this one. In the early 1900's Powers showed that Mersenne had also missed the primes 289 - 1 and 2107 - 1.

By 1947 Mersenne's range, p < 258, had been completely checked and it was determined that the correct list is: p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.

Many theorems have been formed using Mersenne's information. Nowadays, there are a lot of groups in the world that are determined to find the next Mersenne prime. One of the most well known and successful of these groups in called the GIMPS project. GIMPS stands for Great Internet Mersenne Prime Search. GIMPS proved 2756839 - 1 is the 32nd (January 15, 1997), 2859433 - 1 is the 33rd (March 28, 1997), 21257787 - 1 is the 34th (August 28, 1997) and 21398269 - 1 is the 35th (October 11, 1997) by completing the tests for the remaining smaller exponents. All exponents less than 1,481,800 have now been tested at least once.

The largest of the 38 now known primes is 26972593.
The largest known prime is usually a mersenne prime.

After the 23rd Mersenne prime was found at the University of Illinois, the mathematics department was so proud that they had their postage meter changed to stamp "211213 - 1 is prime" on each envelope. Mr. Calkins and his students are also having a share in finding the next Mersenne prime through the GIMPS project - the 39th. If you would like to see the latest Mersenne prime, go to the URL address: ftp://entropia.com/gimps/prime4.txt This number is a staggering 2,098,960 digits long!

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Sources:

This project was prepared and presented by students Kaarsten Harris and Jamos Mitchell in 1998.
This project was updated and presented by Nathan Lippi and Jim Frey