a prime primer
DESCRIPTION
A talk (in progress) on primes and primality testsTRANSCRIPT
Prime Time
December 2008
Morten Jagd Christensen
Thrane & Thrane A/S
is
25288891392479400456592804986000222944998316082937128866378556017873124536111976520731519116376124029996303932610315220549847432414563643
prime?
313-05-2009Skriv titel på præsentation
Agenda
Number theoryProof of infinite number of primesDistribution of primesFermat s little theorem
Algorithms for primality testing
Testing small primes
Using your graphics card for fast calculationsOpenGL graphics accelerationOpenGL Shading Language programmable graphics pipeline
Testing large primesFast multiplications
GNU Multi Precision library
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What is a prime number?
A number which is only divisible by it self and 12 is the only even prime numberOdd prime numbers end in 1, 3, 7 or 9
All natural numbers are uniquely described by prime factors
4=22 6=2 2 8=23 ... 2100 = 22 3 52 7 etc.can be found by factoring into primes
Consider the positive integers 1,2,3,4,.....Primes behave like statistical fluctuations taking up as muchroom as possible considering the requirement of a completesequence of integers
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How many primes are there?
Euclid (300 BC) proved there are infinitely many primes
Construct the following new number
m= (2 3 5 7 11 13 17... n) + 1
Neither of 2..n divides m
-> There must be at least one prime larger than n thatdivides m
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Mother of prime algorithms
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Mother of prime algorithms
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Mother of prime algorithms
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Mother of prime algorithms
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Mother of prime algorithms
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Fermat s little theorem
Composite testif a < p
)1( p
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Fermat s little theorem
Composite testif a < p
so (p,a) = (503, 2)2^502 = 1 (mod 503) could be prime
and (p,a) = (501,2)2^501 = 4 (mod 500) is composite
)1( p
By the way25288891392479400456592804986000222944998316082937128866378556017873124536111976520731519116376124029996303932610315220549847432414563643
is prime
1413-05-2009Skriv titel på præsentation
References
D.E. Knuth The art of Computer Programming
Numerical Recipes in C++
GMP Manual
OpenGL Orange Book
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