1 / 24

The Primal Enigma of Mersenne Primes

The Primal Enigma of Mersenne Primes . Are Mersenne Primes finite or infinite?. By Irma Sheryl Crespo Proof and its examples, puzzle, and the overall presentation by Irma Crespo. 44th Mersenne Prime Found!. ISCrespo 2008. NEWSFLASH!!!. ISCRESPO. ISCrespo 2008.

harmon
Download Presentation

The Primal Enigma of Mersenne Primes

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Primal Enigma of Mersenne Primes Are Mersenne Primes finite or infinite? By Irma Sheryl Crespo Proof and its examples, puzzle, and the overall presentation by Irma Crespo.

  2. 44th Mersenne Prime Found! ISCrespo 2008 NEWSFLASH!!! ISCRESPO

  3. ISCrespo 2008 44th Mersenne Prime Discovered • The new prime number is M32582657 or 232,582,657-1 with 9,808,358 digits. Nearly close to the coveted 10 million digits that has a $100,000 reward attach to it. • Mersenne Primes are named from the digit to which two is raised. Therefore, the recently discovered number is M32582657 because two is raised to the power of 32,582,657. • This newest member of Mersenne Primes was generated on September 4, 2006 by the Central Missouri State University team led by Dr. Cooper and Dr. Boone. ISCRESPO

  4. ISCrespo 2008 But wait! What are Mersenne Primes??? Let's turn back time... ISCRESPO

  5. ISCrespo 2008 All About Father Mersenne • A 17th century French mathematician and physicist. • Jesuit educated and friar of the Order of Minims. • Known as an effective clearinghouse of scientific information. • He conjectured that Mp of Mn=2p-1 was prime for p=2,3,5,7,13,17,19,31,67,127, and 257 and composite for all other primes p  257. This was subjected to challenges. • Despite the intensive scrutiny on the above conjecture, his name was still attributed to Mersenne Primes. • Mersenne Primes refer to prime numbers that are found by raising 2 to a certain power and subtracting one from the total ( 2p-1). 1588-1648 ISCRESPO

  6. ISCrespo 2008 5 Errors on Mersenne’s Conjecture • Pervusin (1883) and Seelhoff (1886) proved independently that M61 was prime. (p=61 was not on Mersenne’s list) • Cole (1903) discovered factors for M67. It was composite. (p=67 was on Mersenne’s list for prime exponents) • Powers (1911) found M89 was prime. (p=89 was not on Mersenne’s list) • Fauquembergue and Powers (1914-1917) proved independently that M107 was prime. (p=107 was not on Mersenne’s list) • Kraitchik (1922) discovered that M257 was composite. (p=257 was on Mersenne’s list for prime exponents) ISCRESPO

  7. ISCrespo 2008 Errors Corrected • By 1947 Mersenne's range, p < 257, 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. ISCRESPO

  8. ISCrespo 2008 A Glance at Perfect NumbersPerfecting the Imperfect • Mystical Property: There is divinity in perfect numbers. • Definition: A positive integer n is said to be perfect if n is equal to the sum of all its positive divisors, not including n itself.Ex. 6 = 1 + 2 + 3 • Theorem: If 2k-1 is prime (k > 1), then n = 2k– 1(2k – 1) is a perfect number. Ex. If k = 3, then, 23-1=7, which is a prime. So, n = 23– 1(23 – 1) = 22(7) =4•7=28: a perfect number. • Lemma:If n = p1k1p2k2…prkr is the prime factorization of the integer n >1, then the positive divisors of n are precisely those integers d of the form d = p1a1p2a2…prar, where 0 ai  ki(i = 1,2,…r). Ex. If n = 180, then, 180 = 22 •32 •51. So, the positive divisors of 180 are integers of the form 2a1•3a2 •5a3 where a1 = 0,1,2; a2 = 0,1,2; and a3 =0,1 because 22 is 20,21,22; 32 is 30,31,32; and 51 is 50,51. ISCRESPO

  9. ISCrespo 2008 Perfect Numbers and Mersenne PrimesRelatively Related • To recap, we’ll set k=p so that the perfect numbers are defined by 2p– 1(2p – 1) while Mersenne Primes are defined by 2p – 1. • It is apparent from the given formulas that a known perfect number can be generated from a Mersenne prime. • If 2p – 1 describes a Mersenne prime, then the corresponding perfect number is equal to 2p– 1(2p – 1) where p=2,3,5,7,13,17,19,31 or other Mersenne exponents. • Obviously, 2p– 1(2p – 1) will always be a perfect number whenever 2p – 1 is a prime number. • Contrary to primes, all perfect numbers are even and all perfect numbers end in 6 and 8 alternately. ISCRESPO

  10. ISCrespo 2008 Where’s the proof? • If M is a prime number equal to 2p – 1, then we have to find the factors for 2p– 1(2p – 1) or 2p– 1 * M. • The factors of 2p– 1 are 1, 2, 4, 8, 16, 32, 64, 128 ... up to 2p-3, 2p-2 and 2p-1. • The rest of the factors of the perfect number, 2p– 1(2p – 1), are each of the already found factors multiplied by 2p– 1. • The sum of the factors of 2p-1 is 1 *( 2p-1) . • The sum of the factors of the perfect number (except the factor of the number itself) is (2p-1- 1) (2p – 1). • Adding both sums will result to 2p– 1(2p – 1), the perfect number. • Thus, 2p– 1(2p – 1) is a perfect number whenever 2p – 1 is a Mersenne prime. ISCRESPO

  11. ISCrespo 2008 Table of the Factors Adding the Sums + The Perfect Number ISCRESPO

  12. ISCrespo 2008 Example please… • If M = 2p – 1 and p=3 (one of Mersenne’s prime exponents) then, 23 – 1= 8 – 1=7, which is a prime number. • The factors of 2p– 1=23– 1= 22 = 4 are 1,2, and 4. • The factors of 2p– 1(2p – 1)= 23– 1(23 – 1) are 1M, 2M and 4M or 1(7),2(7), and 4(7). • Adding the factors of 2p– 1 = 23– 1: 1+2+4 = 7 = M, a Mersenne Prime. • Adding the factors of the perfect number 23– 1(23 – 1) except 4M or 4(7), we have 1(7) + 2(7) = 21. • Putting together the sum of the factors of 23– 1 and 23– 1(23 – 1): 7+21 = 28 = 23– 1(23 – 1), which is the perfect number. ISCRESPO

  13. ISCrespo 2008 Prime or Composite?That is the question. Fast Forward to the Present! ISCRESPO

  14. ISCrespo 2008 The Lucas-Lehmer Test ISCRESPO

  15. ISCrespo 2008 … Lucas-Lehmer Code Made Simple by Wikipedia ISCRESPO

  16. ISCrespo 2008 Go over…just checking. Do you still remember? ISCRESPO

  17. ISCrespo 2008 Search for Mersenne’s P  257 ISCRESPO

  18. ISCrespo 2008 Euclid’s Even Perfect : 2p– 1(2p – 1) Hint: Take the first four of Mersenne’s prime exponents & plug into the “perfect” formula. ISCRESPO

  19. ISCrespo 2008 A Number Neither Prime Nor Composite ISCRESPO

  20. ISCrespo 2008 The Final Answers ISCRESPO

  21. ISCrespo 2008 Last Conundrum Since the 44th Mersenne Prime is found, can we obtain the largest perfect number from it? How many known perfect numbers are there? Something to think about. It’s already out there and it’s close to 20 million digits! ISCRESPO

  22. ISCrespo 2008 The Quest for Primes Continues! ISCRESPO

  23. ISCrespo 2008 The Primal Enigma of Mersenne Primes Are Mersenne Primes finite or infinite? ISCRESPO

  24. ISCrespo 2008 References BOOKS • Burton, David M., History of Mathematics.New York:McGraw- Hill,2007. • Fraleigh,John B., A First Course in Abstract Algebra. Boston: Pearson Education, 2003. WEBSITES • www.drmath.com • www.mathforum.com • www.mersenne.org • www.mersenneforum.org • www.wikipedia.com • www.wolfram.com ISCRESPO

More Related