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Final Math Paper Final Math Paper James Marden December 3, 2016 Introduction Mersenne primes are a set of primes which are of the form: (2p) − 1 = M There are currently only 49 known Mersenne primes, the smallest of which is 22 − 1 = 3 , and the largest being (274207281) − 1. Of these, 15 have been found through the Great Internet Mersenne Prime Search (GIMPS). None of the Mersenne primes were actually discovered by Mersenne himself. Marin Mersenne Marin Mersenne was a French music theorist, mathematician, and thinker during the 1600's. He was said to be "the center of the world of science and mathematics during the first half of the 1600s." In the wake of the Renaissance, pure mathematics continued to flourish in Paris in particular. He was an ordained priest and professor of theology and philosophy at Nevers. He would go on to work with the likes of Descartes, Pascal, and Galileo as well as almost 140 others. Mersenne was born to a working class French family in 1588. His family struggled economically, but was able to make ends meet to send him the Collge du Mans until he was sixteen, and after that the newly founded Jesuit School in La Flche. The Jesuit school was set to be a model school for 1 students of all financial backgrounds. This is likely why Mersenne was able to attend the school without issue. Had Mersenne not been born during the Renaissance and the push for affordable schooling he likely would not have been able to attend a high caliber school and go on to do the work he did. After completing his time at the Jesuit School, his father pushed him to- wards a career in the church. He spent several years studying theology and philosophy at the Sorbonne. He would receive a degree in Philosophy, but gained an appreciation for theology during his time there. Mersenne resisted studying religion formally at first, but eventually realised he could balance his studies as well as a religious career. When leaving Paris, he stayed at a convent of the Minims. He decided to leave Paris to study at the Order of the Minims. The Minims believed they practiced the least religion on Earth. They lived lives of prayer ans scholarship. They wore simple wool clothes, as shown below, and lived simple lives. He became an ordained priest in 1612 and would begin teaching in 1614. Mersenne's first works were in theology and philosophy. 2 It was not until the late 1620's that Mersenne returned in full to Mathematics. He would go on to organize meetings of scholars to review scientific and mathematical papers from France and around the world. Although he did not publish much in mathematics, he published papers in acoustics and the speed of sound as well as being a facilitator for the flow of mathematical and philosophical ideas in Europe. Most of Mersenne's notable work was in harmonics, acoustics, and music the- ory. In the 1630's Mersenne became ill and began travelling in an attempt to cure his illness. Athough unsuccessful in finding a cure, he did meet many mathematicians across Europe, mostly in Holland. During the late 1630's and early 1640's, Mersenne's advice and influence in Bordeaux and Guyenne would lead to the formation of the Royal Academy of Sciences(Acadmie Royale des Sciences). He would undergo an unsuccessful surgery to remove an abscess in his lung. Mersenne died shortly after. Notable Types of Prime Numbers Fermat Primes: Primes of the form 22n + 1 (these will be discussed in detail 3 later) Mersenne Primes: Primes of the form 2p − 1 Double Mersenne Primes: Primes of the form 22p−1 − 1 or 2Mp − 1 Wagstaff Primes: Primes of the form (2p + 1)=3. A larger amount of prime numbers produce Wagstaff Primes, compared to Mersenne Primes, but the results are far more difficult to test and are not nearly as large. The largest Wagstaff Prime has just four million digits, compared to the largest Mersenne Prime which contains 22 million. Proth Primes: Primes of the form k ∗ 2n + 1, these are further classified into Cullen Primes which are of the form n ∗ 2n + 1. The largest known Proth Prime contains over nine million digits, while the largest Cullen Prime is just over two million digits. Sophie Germain Primes: Primes which form another prime when used in the equation 2p + 1. It is conjectured that there are an infinite number of Germain primes, but it has not been proven in the positive or negative. Notable Mention: Twin: p, p + 2 Cousin: p, p + 4 Sexy: p + 6 Fermat Primes Fermat Primes, as mentioned, are primes of the form 22n + 1. They are only important due to historical context and that they lead into Fermat-Mersenne Primes. Fermat primes are an odd case. The first five Fermat Numbers are all prime (n=0 to n=4), but none have been discovered beyond that. All numbers tested from n=5 and above are composite. This is an incredibly odd and interesting pattern. The formula works so well to start, but is entirely in ruins after just a few repetitions. It is known that not all Fermat numbers are prime, but it is unknown how many Fermat Primes there are. 4 Mersenne-Fermat Primes Mersenne-Fermat numbers are those of the form (2pr − 1)=(2pr−1 − 1). When p=2, these are also Fermat numbers. The denominator of the expression is the form of a Double Mersenne Prime. There are only eight Mersenne- Fermat Primes, the largest of which being MF(59,2) such that MF (p; r) = (2pr − 1)=(2pr−1 − 1) This produces: Although a large number that will cause most calculators to produce "In- finity," it is relatively small by comparison. It only has 18 digits while the largest Mersenne Prime has more than a million times as many. Mersenne Primes are a special class of numbers and more specifically of prime 5 numbers. Not all primes are Mersenne Primes, and not all Mersenne numbers are Prime. The first non-prime Mersenne Number is p = 11, producing 211 − 1 = 2047 which is divisible by 23 and 89. The prime numbers which produce Mersenne Primes are increasingly spread out over the following series of primes. Proof of Why the Exponent Must be Prime If: (2n) − 1 = M produces a prime number, M, then n must also be prime. We will prove this by contradiction. Let us assume that n may be composite, n = x ∗ y. In this case 2xy − 1 then, 2xy − 1 = 2x − 1 ∗ (2xy − 1 + 2xy − 2 + ::: + 2x + 1) This means that 2x − 1 divides 2xy − 1 and if one were to show the opposite proof it would be shown that 2y − 1 also divides 2xy − 1. Therefore, there cannot be any numbers x and y that multiply to n if the Mersenne number is represented in the form 2n − 1 so n must be a prime number p. https://primes.utm.edu/notes/proofs/Theorem2.html Lucas-Lehmer Test The Lucas-Lehmer test is a primality test specific for Mersenne Numbers. It is incredibly fast compared to other primality tests, allowing large Mersenne Numbers to be tested relatively quickly. The test says that to show a Mersenne Number is prime, it must be shown that s(n) = (s(n−1)2−2)Modp where s(0) = 4 and s(n) = s(n − 1)2 − 2. The first few numbers of this se- quence are 4, 14, 194, 37634... This is the test used by GIMPS and most other programs that attempt to find Mersenne Primes. Great Internet Mersenne Prime Search The Great Internet Mersenne Prime Search was started in 1996, finding their first Mersenne Prime later that year. Since, they have discovered 14 more, the most recent being in January of 2016. The idea of the project is fairly simple, but incredibly effective on a large scale. The project asks people to 6 use the free CPU on their computers to help run primality tests and check factorizations. The Website allows people to form teams or work individually to donate CPU time to help check potential primes. To join in the search, users simply regis- ter an account, download the program Prime95 and allow their processor to run the program and check primes with unused processor cycles. The project will award $3,000 to any user who discovers a Mersenne Prime with less than 100,000,000 digits, and $50,000 to a user who discovers a Mersenne Prime with more than 100,000,000 digits. The most recently discovered Mersenne Prime (January 7th 2016) is the largest known prime number. The six largest prime numbers are all Mersenne Primes, with 15 of the 16 largest prime numbers being Mersenne Primes. This is due to the fact that the formula to produce new Mersenne numbers is very straightforward and more importantly that the Lucas-Lehmer test is incredibly fast compared to other known primality tests for other forms of prime numbers. 7 Refernces 1. http://www.mersenne.org/ 2. http://primes.utm.edu/mersenne/ 3. http://planetmath.org/proofoflucaslehmerprimalitytest 4. http://fermatslibrary.com/s/a-really-trivial-proof-of-the-lucas-lehmer-test 5. http://mathworld.wolfram.com/Lucas-LehmerTest.html 6. http://oeis.org/A003010 7. http://www-history.mcs.st-and.ac.uk/Biographies/Mersenne.html 8. https://primes.utm.edu/mersenne/LukeMirror/lit/lit_024s.htm 9. http://mathworld.wolfram.com/MersennePrime.html 10. http://www.mersenne.org/primes/ 11. http://www.mersenne.org/gettingstarted/ 12. http://www.math.wichita.edu/history/men/mersenne.html 8.
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