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

NEWSFLASH!!! ISCrespo 2008 ISCRESPO

44 th Mersenne Prime Discovered The new prime number is M or 2 32,582, 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 M 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 2008 ISCRESPO

But wait! What are Mersenne Primes??? ISCrespo 2008 ISCRESPO

All About Father Mersenne A 17 th 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 M p of M n =2 p -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 ( 2 p -1) ISCrespo 2008 ISCRESPO

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

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 2008 ISCRESPO

A Glance at Perfect Numbers Perfecting 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 = Theorem: If 2 k -1 is prime (k > 1), then n = 2 k – 1 (2 k – 1) is a perfect number. Ex. If k = 3, then, =7, which is a prime. So, n = 2 3 – 1 (2 3 – 1) = 2 2 (7) =47=28: a perfect number. Lemma: If n = p 1 k 1 p 2 k 2 …p r k r is the prime factorization of the integer n >1, then the positive divisors of n are precisely those integers d of the form d = p 1 a 1 p 2 a 2 …p r a r, where 0  a i  k i (i = 1,2,…r). Ex. If n = 180, then, 180 = So, the positive divisors of 180 are integers of the form 2 a 1 3 a 2 5 a 3 where a 1 = 0,1,2; a 2 = 0,1,2; and a 3 =0,1 because 2 2 is 2 0,2 1,2 2 ; 3 2 is 3 0,3 1,3 2 ; and 5 1 is 5 0,5 1. ISCrespo 2008 ISCRESPO

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

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

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

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

Prime or Composite? That is the question. ISCrespo 2008 ISCRESPO

The Lucas-Lehmer Test ISCrespo 2008 ISCRESPO

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

Go over…just checking. ISCrespo 2008 ISCRESPO

Search for Mersenne’s P  ISCrespo 2008 ISCRESPO

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

A Number Neither Prime Nor Composite ISCrespo 2008 ISCRESPO

The Final Answers ISCrespo 2008 ISCRESPO

Last Conundrum Since the 44 th 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 2008 ISCRESPO

The Quest for Primes Continues! ISCrespo 2008 ISCRESPO

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

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, WEBSITES ISCrespo 2008 ISCRESPO