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Mersenne PrimesHow to generate more prime numbers?Mersenne (1588-1648) generated primes using formula: where p is a prime

M2 = 3; M3 = 7, M5 = 31, M7 = 127Is 127 prime? Refer to list of primes (www.utm.edu/research/primes) M11 = 2047 = 23 89 (not prime)

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Mersenne PrimesMp 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 primesThe search for largest known prime…

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Mersenne PrimesM43,112,609 is the largest prime found on 23 Aug 2008 by Edson SmithIt contains 12,978,189 digits!If one newspaper page contains about 30 000 digits, then M43,112,609 will occupy 433 newspaper pagesFile (17 MB): 45th Mersenne PrimeWhat are the first and last digits?

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Mersenne PrimesHow 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 EnrichmentsSearch the Internet for information on:

Goldbach’s ConjectureFermat Primes and NumbersConstruction 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)