What is a Mersenne Prime

1
What is a Mersenne Prime?

• Who was Mersenne?
• What is a perfect number?
• What is the largest/ most recently discovered
  Mersenne Prime?
• How can I be a part of history?

2
What is a Mersenne Prime?

• A prime that is one less than a power of 2
• MP 2 ? - 1 where m is
  prime
• Why is a Mersenne Prime special?
• Mersenne Primes are special because they are
  required to find perfect number.
• What is a perfect number??

3
What is a perfect number?

• First appeared in the 4th century BCE in Euclids
  Elements
• A perfect number is a number whose proper
  divisors (all factors including 1, but excluding
  the number itself) sum to equal the number.
• Modernized to If, for some m gt 1, 2 ? - 1 is
  prime
• then 2 ? -1 ( 2 ? - 1)
  is a perfect number.
• EX 6 123 28 124714
• It is unknown if there are any odd perfect
  numbers and if the set of perfect numbers is
  infinite.

4
Who was Mersenne?

• Marin Mersenne (1588-1648)
• Minimite friar, close friend of Galileo, Fermat
  and Descartes
• Mersenne was the recorder/secretary of a group of
  scientist and mathematicians
• His widespread correspondence of his groups
  activities contributed to a sense of unity in
  mathematical interest
• Called the Walking Scientific Journal
• Mersenne compiled a list with exponents up to 257
  BUT erroneously included M67 and M257 and did not
  include M61, M89, and M107. There was no
  indication as to how they were found.

5
History of Mersenne Primes

• The first 4 perfect numbers, and therefore
  Mersenne Primes, are known in antiquity. No
  documents exist to show their discovery.
• The 5th Mersenne Prime was published anonymously
  in 1456
• They were not discovered in order- some were
  overlooked and some were included but later found
  to be composite numbers.
• 7 more Mersenne Primes were found before

6
Computers!!!

• 1952- First MP discovered by computer after a 38
  year lapse. M521 (13) at UCLA at the US National
  Bureau of Statistics.
• 1963- The 21st-23rd MPs were found while a
  professor at Univ. of Illinois was testing a
  supercomputer.
• 1978- 2 High school students made
• national news and the front page of
• the NY times when the computer they
• volunteered located the 25th MP.
• The 27th- 34th MPs (except 29th) were discovered
  by Slowinski- a Cray Computer Laboratories
  employee that ran the program in his computers
  spare time, without permission, between 1979
  and 1996. A reporting delay took place while
  Slowinski sought permission

7
What is GIMPS?

• In late 1995, computer programmer George Woltman
  creates the free download of the Great
  International Mersenne Prime Search (GIMPS)
  program to any individuals who would like to
  volunteer their computers memory to the cause.
• The 35th- 44th MP were discovered using GIMPS.
  GIMPS is running at 29 TFLOPS. (Tera Floating
  Point Operation Per Second) 29 x 1012
  operations per second, as of May 08

8
What is the largest/most recently discovered
Mersenne Prime?

• The 44th MP was discovered September 4, 2006
  by one of 700 clustered GIMPS volunteer computers
  belonging to Central Missouri State University
  professors Dr. Curtis Cooper and Dr. Steven
  Boone. (They found the 43rd too, what are the
  odds of that???)
• The 44th MP is (232,582,657)-1. It has
  9,808,358 digits, just 191, 642 digits short of
  claiming the 100,000 prize offered by the
  Electronic Frontier Foundation.
• A standard word processing program would use
  2,616 pages to display. So here they
  are

9
Just kidding ?
10
How Can I Be a Part of History?

• Each new MP that is discovered is attributed to
  the individual whose computer it popper up on
  (along with Woltman, the program designer)
• Of the 10 MP that were found by GIMPS, 5 were
  discovered by undergrad college students and
  professors
• In addition to being the newest MP discoverer,
  you will be awarded 50,000 and 25,000 will be
  donated to charity in your name!
• WOW! Rich AND Famous!!!!
• Snap your lazy, idling computer into shape by
  downloading the program at www.mersenne.org

11
Resources

• http//www.mersenne.org/ Retrieved 7/15/08
• http//primes.utm.edu Retrieved 7/15/08
• A History of Mathematics, An Introduction- Victor
  Katz, 1993
• A History of Mathematics- Second Edition- Carl
  Boyer, 1968
• http//en.wikipedia.org/wiki/Mersenne_primes
  Retrieved 7/9/08