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Mersenne Primes

Joseph Fryer and Casey Detro, Miami University

Marin Mersenne’s mathematical contributions helped paved the way for advances in mathematics. In this article, his life is not only depicted but his mathematical intellect is highlighted through his achievements. Since his time, technology has helped further the fi nding of Mersenne primes. Prime numbers are large concepts for students to learn in school. Mersenne primes further examine the study of prime numbers.

Introduction Prime numbers have intrigued the minds of mathematicians throughout history. Prime numbers are characterized by only having two factors: 1 and itself. Th e development of these numbers was a major advancement in number theory. Whether mathematicians studied these numbers in their free time Many people or they studied them to make advances in only know of number theory, many positive contributions Mersenne for have come from such investigations. One his famous such mathematician who made a great advance in the study of prime numbers was conjecture for Marin Mersenne. He is most widely known which numbers for a certain kind of prime number, Mersenne (Image Courtesy of http://fr.academic.ru/dic.nsf/frwiki/1240555 are named for prime numbers. Fig 1 Marin Mersenne (1588-1648) him; however, Mersenne primes are a special type of prime number. Th ey have distinct characteristics in shrewd thinker, a brilliant experimentalist, he was also comparison to “regular” prime numbers. Th e an inferior mathematician” while some even an important mathematical concept of Mersenne primes believe he needed psychiatric evaluation infl uence was developed in the search for perfect (Buske & Keith, 2000). Although, many on young numbers. A perfect number is a number in people only know of Mersenne for his which the sum of its proper factors equals the famous conjecture for which numbers are mathematicians original number. named for him, he was also an important infl uence on young mathematicians. He For example, 6 is a perfect number. The factors became a mentor to young researchers in the of 6 are: 1,2,3,6. When we add the factors fi elds of mathematics and science. Mersenne together that are less than 6 we get: 1+2+3=6 was discouraged because these fi elds were in decline, unable to attract a suffi cient number History of scholars. He created a research center in Marin Mersenne was born in 1588 in Europe and used it to keep in contact with France. He studied at the Jesuit Collége other mathematicians and scientists and du Mains of La Fléche (Wanko, 2005) notify them of any recent discoveries in their and became an ordained priest in 1614. fi elds. He was a mentor to a young Pascal. Mersenne is described in many diff erent Without such mentoring, it is hard to say ways by historians. Some believe he was “a if we would have the same theorems we do

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today. Mersenne was friends with other a challenge to fi nd more perfect numbers. famous mathematicians such as Descartes Euclid wrote a series of books entitled and Fermat. Unlike Mersenne, they did not Elements. In Proposition 36 of book IX of enjoy reading the works of other scholars the Elements, Euclid states a formula to fi nd because they feared they would be overly perfect numbers. infl uenced by their discoveries. “Mersenne Euclid’s formula stated: felt that much more could be accomplished If 2k − 1 is prime for k>1, then through shared ideas and communication n =2k − 1(2k − 1) is a perfect number. and thus took it upon himself to correspond For example if k =2 then 22 − 1=3 with young mathematicians and scientists and (which is prime). Now we can plug 3 into provide them with detail of others’ fi ndings” Euclid’s equation. n =22− 1(3) = 6, (Wanko, 2005). After Mersenne had passed which is a perfect number. away in 1648, it was determined that he had seventy eight diff erent correspondents all How does Euclid’s formula for perfect across Europe (Wanko, 2005). Mersenne’s numbers relate to the primes that Mersenne discovered? “If Euclid’s theorem were During famous conjecture came in 1644, only four years prior to his death. true, the problem of fi nding larger perfect his time, numbers could be boiled down to fi nding 2k − 1 Mersenne Brief explanation of primes values for k where the expression had to Before we delve further into Mersenne is prime,” (Wanko, 2005). Th is is when Marin Mersenne stepped in and formed his compute all primes, it is helpful to fi rst provide a brief overview of prime numbers. Recall that a conjecture. He was also interested in fi nding calculations more perfect numbers, so he took it on himself prime number only has two factors: one k to fi nd values of to satisfy Euclid’s formula. by hand. and the number itself. For example, fi ve is a 2k − 1 Mersenne was aware that in order for He spent prime number because it only has 2 factors, 1 k and 5. If a number has more than these two to be prime, itself must also be prime. In countless 1644, he claimed that he had found all the factors it is said to be composite. Primes are 2k − 1 hours trying considered a building block in mathematics. prime numbers that make prime. to fi nd As the Fundamental Th eorem of Algebra His conjecture consisted of the numbers: 2, values of k. indicates, all integers may be represented as a 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257. unique product of primes. Prime numbers are Th ese numbers then became known as the Mersenne Primes. important in mathematics, and it is crucial k for students to learn about them while in Mersenne’s list of possible values for was middle school. questioned by some in regards to his degree of accuracy. During his time, Mersenne had Mersenne Primes to compute all calculations by hand. He spent countless hours trying to fi nd values of Th e discovery of Mersenne primes can be k attributed to contributions made by Euclid, . Since they were all worked out by hand, the Pythagoreans, and Marin Mersenne there were probably some computational himself. Th e history begins with an increasing errors. “It wasn’t until 1947p that all of the interest in fi nding perfect numbers. possible Mersenne primes ( ) were checked During Mersenne’s time, the Pythagoreans using computers,” (Wanko, 2005). Much had only found four perfect numbers: 6, to the surprise of many mathematicians, 28, 496, and 8128. Th e fact that only four his conjecture was correct (although not perfect numbers had been found presented complete).

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Mersenne’s original conjecture: from all the people participating. Anyone can M =2k − 1 is prime for k = 2, 3, 5, 7, 13, help participate in the search if they have the 17, 19, 31, 67, 127, and 257. right type of computer. Monetary rewards are even being off ered to whoever fi nds the Actual Primes (Wanko, 2005): With k ≤ 257, k fi rst ten-million-digit prime number. In M =2 − 1 is prime for k = 2, 3, 5, 7, 13, 1999, the search found its then record high 26972593 − 1. 17, 19, 31, 67, 127, and 257. prime It was found by Nayan Hajratwala of Michigan using the GIMPS One may notice that this list of primes software that he downloaded on his computer starts out by listing the prime numbers in 11 (Buske & Keith, 2000).Th is number is over order, but isn’t also a prime number? two million digits long and could fi ll over Mersenne noticed that not every prime seven miles of digits and commas (Buske & number satisfi ed the equation. For instance, 11 211 − 1 = 2048. Keith, 2000). Th en, in 2004, another prime if we try to use we get: 2048 became the largest known world prime, We know that can’t be prime because it 224036583 − 1, 2 consisting of over seven is divisible by (2 × 1024).

Now, with the aid of technology, people continue to search for larger and larger prime and Mersenne (Image Courtesy of gilchrist.ca/jeff/MprimeDigits/index.html) Fig 2 Beware of false generalizations! Fig 3 Mersenne digits calculator prime numbers. Mersenne Primes and Technology million digits (Wanko, 2005). At the time In the seventeenth century, when Mersenne of this writing, the largest known Mersenne 43112609 made his famous conjecture, the only method prime is 2 − 1. To date there are 47 mathematicians had for fi nding large numbers known Mersenne primes and the search is was by hand. Now, with the aid of technology, still ongoing and it is surprising at how large people continue to search for larger and larger these numbers can be. prime and Mersenne prime numbers. “In 1996, George Woltman, using networking software Conclusion by Scott Kurowski, created the database In conclusion, Marin Mersenne made a GIMPS (Great Internet Mersenne Prime signifi cant contribution to mathematics. His Search), which coordinates the eff orts of more intellect and mathematical curiosity about than 8,000 computer users internationally in perfect numbers helped to pave the way a practice known as distributed internet data for his conjecture. Today, with the help of processing” (Buske & Keith, 2000). Th e main computers, mathematicians have found the server then distributes and collects information largest prime number to date through the

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use of Mersenne primes. Maybe in the near CASEY DETRO is future the next largest prime number will be currently a senior found. After all, there is an infi nite number at Miami University of them! majoring in Middle Childhood Education Works Cited in the areas of math Buske, D, & Keith, S. (2000). GIMPS fi nds and science. She another prime. Math Horizons, 7(4), 19- enjoys reading, working out, and 21. spending time with friends and family. Mersenne Prime # of Digits Calculator. (2008). In Gilchrist online. Retrieved from http://gilchrist.ca/jeff/MprimeDigits/ index.html O’Connor, J. J., Robertson, E. F. (2009). JOSEPH FRYER is Perfect Numbers. Retrieved from a senior at Miami http://www-history.mcs.st-and.ac.uk/ University. He is HistTopics/Perfect_numbers.html currently majoring Prime Number. In Nombre Premier online. in Middle Childhood Retrieved from http://fr.academic.ru/dic. Education in the nsf/frwiki/1240555 areas of math and Wanko, J. (2005). Th e Legacy of Marin science. He enjoys camping and Mersenne: Th e Search for Primal Orders spending time with his friends and and the Mentoring of Young Minds. family. Mathematics Teacher. Reston, VA: Th e National Council of Teachers of Mathematics.

Behind the scenes

“When an idea is served up from behind the scenes [unconsciousness], your neural circuitry has been working on it for hours or days or years, consolidating information and trying out new combinations.”

Eagleman, D. M. (2011). Incognito: Th e secret lives of the brain, 7. Pantheon Books, NY.

“Insanity is doing the same thing, over and over again, but expecting different results.”

- Albert Einstein

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