The Primal Enigma of Mersenne Primes The Primal Enigma of Mersenne Primes

By Irma Sheryl CrespoProof and its examples, puzzle, and the overall

presentation by Irma Crespo.

Are Mersenne Primes finite or infinite?

NEWSFLASH!!!

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4444thth Mersenne Prime Discovered 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.

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But wait! But wait! What are Mersenne Primes??? What are Mersenne Primes???

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All About Father MersenneAll 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

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5 Errors on Mersenne’s Conjecture5 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)

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Errors CorrectedErrors 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.

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A Glance at Perfect NumbersA Glance at Perfect NumbersPerfecting the ImperfectPerfecting 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 = p1k1p2

k2…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 = p1a1p2

a2…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.

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Perfect Numbers and Mersenne PrimesPerfect Numbers and Mersenne PrimesRelatively RelatedRelatively 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.

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Where’s the proof?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.

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Table of the FactorsTable of the Factors

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2p-1 2p – 1(2p – 1)

1 1(2p – 1)

2 2(2p – 1)

4 4(2p – 1)

8 8(2p – 1). .. .. .

2p-3 2p-3(2p – 1)

2p-2 2p-2 (2p – 1)

2p-1 2p-1 (2p – 1)

SUM 1 *( 2p-1) (2p-1- 1) (2p – 1)

Adding the SumsAdding the Sums

+ 1 * ( 2p– 1)

(2p-1- 1) * (2p – 1)

2p-1 * (2p – 1)

The Perfect Number

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Example please…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.

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Prime or Composite?Prime or Composite?That is the question.That is the question.

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The Lucas-Lehmer TestThe Lucas-Lehmer Test

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Lucas-Lehmer Code Made Simple Lucas-Lehmer Code Made Simple byby Wikipedia Wikipedia

…

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Go over…just checking.Go over…just checking.

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Search for Mersenne’s P Search for Mersenne’s P 257 257

900 5 89 105 127 7 1952 605 19 3228

40 10 1 1456 2 100 9 21 6 125

57 28 17 126 257 49 8128 55 61 8888

1876 31 111 333 23 7036 520 258 496 15

92 107 3 14 81 29 2201 112 13 96

900 5 89 105 127 7 1952 605 19 3228

40 10 1 1456 2 100 9 21 6 125

57 28 17 126 257 49 8128 55 61 8888

1876 31 111 333 23 7036 520 258 496 15

92 107 3 14 81 29 2201 112 13 96

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900 5 89 105 127 7 1952 605 19 3228

40 10 1 1456 2 100 9 21 6 125

57 28 17 126 257 49 8128 55 61 8888

1876 31 111 333 23 7036 520 258 496 15

92 107 3 14 81 29 2201 112 13 96

900 5 89 105 127 7 1952 605 19 3228

40 10 1 1456 2 100 9 21 6 125

57 28 17 126 257 49 8128 55 61 8888

1876 31 111 333 23 7036 520 258 496 15

92 107 3 14 81 29 2201 112 13 96

Euclid’s Even Perfect : Euclid’s Even Perfect : 2p – 1(2p – 1)

Hint: Take the first four of Mersenne’s prime exponents & plug into the “perfect” formula.

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900 5 89 105 127 7 1952 605 19 3228

40 10 1 1456 2 100 9 21 6 125

57 28 17 126 257 49 8128 55 61 8888

1876 31 111 333 23 7036 520 258 496 15

92 107 3 14 81 29 2201 112 13 96

A Number Neither Prime Nor CompositeA Number Neither Prime Nor Composite

900 5 89 105 127 7 1952 605 19 3228

40 10 1 1456 2 100 9 21 6 125

57 28 17 126 257 49 8128 55 61 8888

1876 31 111 333 23 7036 520 258 496 15

92 107 3 14 81 29 2201 112 13 96

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900 5 89 105 127 7 1952 605 19 3228

40 10 1 1456 2 100 9 21 6 125

57 28 17 126 257 49 8128 55 61 8888

1876 31 111 333 23 7036 520 258 496 15

92 107 3 14 81 29 2201 112 13 96

900 5 89 105 127 7 1952 605 19 3228

40 10 1 1456 2 100 9 21 6 125

57 28 17 126 257 49 8128 55 61 8888

1876 31 111 333 23 7036 520 258 496 15

92 107 3 14 81 29 2201 112 13 96

The Final AnswersThe Final Answers

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Last ConundrumLast Conundrum

Since the 44Since the 44thth Mersenne Prime is found, Mersenne Prime is found, can we obtain the largest perfect number from it? can we obtain the largest perfect number from it?

How many known perfect numbers are there?How many known perfect numbers are there?

Something to think about. Something to think about.

It’s already out there and It’s already out there and it’s close to 20 million digits!it’s close to 20 million digits!

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The Quest for Primes Continues!The Quest for Primes Continues!

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Are Mersenne Primes finite or infinite?

The Primal Enigma of Mersenne Primes The Primal Enigma of Mersenne Primes

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ReferencesReferences

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

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