1 Mersenne Primes How to generate more prime numbers? Mersenne (1588-1648) generated primes using formula: where p is a prime M 2 = 3; M 3 = 7, M 5 = 31,

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1 Mersenne Primes How to generate more prime numbers? Mersenne ( ) generated primes using formula: where p is a prime M 2 = 3; M 3 = 7, M 5 = 31, M 7 = 127 Is 127 prime? Refer to list of primes ( of primeswww.utm.edu/research/primes M 11 = 2047 = 23  89 (not prime)

2 Mersenne Primes M p is prime only for certain primes p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, … M 89 = 6.19  (very big) Try proving M 89 is a prime in the 16 th century without a calculator! How many primes are there? Infinitely many primes The search for largest known prime…

3 Mersenne Primes M 43,112,609 is the largest prime found on 23 Aug 2008 by Edson Smith It contains 12,978,189 digits! If one newspaper page contains about digits, then M 43,112,609 will occupy 433 newspaper pages File (17 MB): 45 th Mersenne Prime45 th Mersenne Prime What are the first and last digits?

4 Mersenne Primes How do people find such big primes? By using a network of computers. How? Method is trial division US$ award for discovery of first 10-million-digit prime (gone!) US$ award for discovery of first 100-million-digit prime

5 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)