Transcript Document

Mersenne Primes
How to generate more prime numbers?
Mersenne (1588-1648) generated
primes using formula: M p  2 p  1
where p is a prime
M2 = 3; M3 = 7, M5 = 31, M7 = 127
Is 127 prime? Refer to list of primes
(www.utm.edu/research/primes)
M11 = 2047 = 23  89 (not prime)
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Mersenne Primes M p  2  1
p
Mp is prime only for certain primes
p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, …
M89 = 6.19  1026 (very big)
Try proving M89 is a prime in the 16th
century without a calculator!
How many primes are there?
Infinitely many primes
The search for largest known prime…
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Mersenne Primes M p  2  1
p
M43,112,609 is the largest prime found on
23 Aug 2008 by Edson Smith
It contains 12,978,189 digits!
If one newspaper page contains about
30 000 digits, then M43,112,609 will
occupy 433 newspaper pages
File (17 MB): 45th Mersenne Prime
What are the first and last digits?
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Mersenne Primes
How do people find such big primes?
By using a network of computers. How?
Method is trial division
www.mersenne.org
US$100 000 award for discovery of
first 10-million-digit prime (gone!)
US$150 000 award for discovery of
first 100-million-digit prime
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Other Enrichments
Search the Internet for information on:
Goldbach’s Conjecture
Fermat Primes and Numbers
Construction of regular polygons using
straightedge and compasses (search for
Carl Gauss): related to Fermat Primes!
Application of number theory (study of
primes) in very secure computer data
encoding (search for RSA cryptosystem)
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