Detecting Primes

1
Detecting Primes

• William Dotson

2
Overview

• Definitions
• How many primes?
• How do we locate?
• How do we prove primality?
• Quick Tests
• Classical Tests
• General Purpose Tests
• Application
• Summary

3
Definitions

• Prime Number an integer p is prime if and only
  if the only positive integers that divide it are
  1 and p.
• Composite Number a integer that is not prime.
• Relatively Prime 2 Integers are relatively
  prime if they share no common factors other 1.
• Pseudoprime a number that can easily be proved
  composite but might not as easily be proved
  prime.
• Mersenne Prime a prime number 2n-1 where n is
  itself prime.

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How Many Primes?

• Answer Infinite (Euclid, 2300 years ago)
• So, How many primes less than x?
• Pi(x) the number of primes less than x
• Pi(3) 2 ,(2,3) Pi(10) 4,(2,3,5,7)
• Prime number Theorem The number of primes not
  exceeding x is asymptotic to xlogx
• Pi(x) xlogx

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How Many Primes? (cont.)

• Actually x / (log(x a)
• a is an arbitrary constant
• 1 is best choice

X Pi(x) x/(log x) x/(log x -1)
1000 168 145 169
10000 1229 1086 1218
100000 9592 8686 9512
1000000 78942 72382 78030
10000000 664579 620420 661459
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Nth Prime

• P(n) nlogn (Hardy Wright)
• P(n) n(log n loglog n 1)(Ribenboim)
• For 1 millionth prime, Hardy Wright gives 13.8
  million, Ribenboim 15.4 million.
• Actually 15,485,863
• P(n) n(log n loglog(n) 1 (loglog(n)
  2)/log n ((loglog(n))2 6loglog(n) 11)/(2
  log2 n) O((log log n / log n)3)(Cipolla,
  Dusart)

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Probability of Being Prime

• Probability of x being prime is 1/(log x)
• E.g. Randomly choose a 1000 digit number. Can
  expect to test log(101000) numbers.
• Can reduce by eliminating certain numbers.

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Locating Primes

• Sieve of Eratosthenes
• Trial Division
• a-PRP
• a-SPRP
• N-1 tests N1 tests
• APR and ECPP
• AKS

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Quick Tests

• For numbers lt 10,000,000,000 most efficient is
  Sieve of Eratosthenes(240 B.C.)
• Start with list of odd numbers.
• Need small primer of known prime numbers

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3
2
1
7
13
11
19
17
25
23
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Quick Tests

• Trial Division Dividing by all prime numbers
  less than the square root of n.
• For 97
• Square root 9.8
• 97/2 48.5
• 97/3 32.3
• 97/5 19.4
• 97/7 13.8
• Next Prime is 11, so stop here. 97 is prime.

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Fermats Theorem a-PRP

• Fermats Little Theorem
• p is a prime, a is any positive integer
• ap a (mod p)
• If p does not divide a, a(p-1) 1(mod p)
• So if a(p-1) mod p ? 1 mod p, p is composite
• If it is equal, then it MIGHT be prime.
• P is called a weak pseudoprime base a, or just
  a-PRP
• The larger the p, the more likely this test
  works.

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Carmichael Numbers

• Obstacle to a-PRP test
• If ap-1 1(mod p) for all a relatively prime to
  p, p is composite.
• Hard to detect
• Rare 2000 less than 109
• Infinitely Many
• 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585,
  15841, 29341, 41041, 46657, 52633, 62745, 63973,
  and 75361.

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a-SPRP

• Improvement
• n-1 2sd where d is odd, s is a non-negative
  integer. N is SPRP if ad 1(mod n) or
  (ad)2r -1 (mod n) for some negative r less than
  s
• ¾ that pass are prime
• Can combine SPRP results to prove primality

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a-SPRP

• Nlt1,373,653 is both 2 and 3 SPRP, then prime
• Nlt25,326,001 is 2,3, and 5 SPRP, then prime
• Nlt118,670,087,467 is 2,3,5,7 SPRP then either
  prime or is the number 3,215,031,751
• Nlt341,550,071,728,321 is 2,3,5,7,11,13,17 SPRP ,
  then prime
• - If the generalized Riemann hypothesis is true
  then if n is a-SPRP for all integers a with
  1ltalt2(logn)2 then n is prime

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Classical Tests

• N-1 test
• N gt 1, If for every prime factor q of n 1 there
  is an integer a such that.
• a(n-1) 1 (mod n) and
• A(n-1)/q is not 1(mod n)
• Then n is prime

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Lucas-Lehmer Test

• Let p be a prime an odd prime.
• The Mersenne number M(p) 2n 1 is prime if and
  only if S(n-2) 0 (mod(M(n)), where S(0) 4 and
  S(k1) S(k)2 2
• Exceptionally fast on computers.
• 2 high school students programmed this theorem in
  1978 discovering the then largest known Mersenne
  Prime number. 221701 1
• GIMPS (Greatest Internet Mersenne Prime Search)
• Latest Found, 224,036,583 1, 7 million digits.
• http//mersenne.org/prime7.txt

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More Classical Tests

• Pocklingtons Theorem
• Pepins Test
• Proths Theorem

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General-Purpose Tests

• Most of the previous tests and theorems are old.
  Date back before most computing.
• 1970s people began using other factors other
  than n1 and began to use n2 1, n2 -1, n2
  n 1, etc etc.
• APR, APR-CL algorithms
• Almost polynomial
• ? (log n)(c2 log log log n)

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Elliptical Curves, ECPP Test

• E(a,b) y2 x3 ax b (with 4a3 27b2 not
  zero)
• E(a,b)/p lies in the interval
  (p1- 2sqrt(p),p12sqrt(p)) and the
  orders are fairly uniformly distributed (as we
  vary a and b).
• Polynomial for some input

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Agrawal, Kayal and Saxena

• First true polynomial algorithm - 2002
• No unproved assumptions
• Deterministic
• O(n6)
• Implementations so far have run around O(n12) but
  this is probably due to large constants for lower
  order members.

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Applications

• Public-Key Cryptography
• Graphics
• Nucleotide Sequencing
• Hash Tables
• Pseudorandom Number Generators
• Natural World

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Mersenne Twister

• 1997 Makoto, Matsumoto
• Period of 219937-1
• Equidistributed
• Faster than most statistically unsound PRNG
• Statistically random in all bits

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Graphics Examples
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Summary

• Prime number theory is old
• Many Methods
• Search will for larger primes will continue
• Computer Scienists and mathematicians will
  continue to look for more efficient methods of
  determining primes.

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References

• M. Agrawal, N. Kayal, N.Saxena. Primes in P.
  Aug, 2002.
• A.O.L. Atkin and F. Morain. "Elliptic curves and
  primality proving," July 1993.
• Wikipedia, the Free Encyclopediahttp//en.wikipedi
  a.org/wiki/Main_Page GFDL. September 2004.
• Aesthetics of Prime Sequencing.
  http//www.2357.a-tu.net/ Turpel Armand 2001.
• Primality Proving. http//www.utm.edu/research/pri
  mes/prove/index.html Chris Caldwell.
• Mersenne Prime Search. http//www.mersenne.org/
  George Woltman. September 15th, 2004.
• Mathworld http//mathworld.wolfram.com/ Wolfram
  Research.