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sci.math FAQ: Prime Numbers


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Archive-name: sci-math-faq/primes
Last-modified: February 20, 1998
Version: 7.5


   
                                 Prime Numbers
                                       
Largest known Mersenne prime

   Mersenne primes are primes of the form 2^p - 1. For 2^p - 1 to be
   prime we must have that p is prime.
   
   2^(2976221) - 1 is prime. It was discovered in 1997.
   
Largest known prime

   The largest known prime is the Mersenne prime described above. The
   largest known non-Mersenne prime, is 391581*2^(216193) - 1, discovered
   by Brown, Noll, Parady, Smith, Smith, and Zarantonello.
   
   Throughout history, the largest known prime has almost always been a
   Mersenne prime; the period between Brown et al's discovery in August
   1989 and Slowinski & Gage's in March 1992 is one of the few
   exceptions.
   
   You can help find more primes. For more information see: The Great
   Internet Mersenne Prime Search home page on http://www.mersenne.org
   
      References
      
   Brown, Noll, Parady, Smith, Smith, and Zarantonello. Letter to the
   editor. American Mathematical Monthly, vol. 97, 1990, p. 214.
   
Largest known twin primes

   The two largest known twin primes are 242206083 * 2^38880 +- 1. with
   11713 digits, found by Indlekofer and Ja'rai in November, 1995.
   
   They are also the first known gigantic twin primes (primes with at
   least 10,000 digits).
   
   Recent record holders are:
   
     * 190116*3003*10^(5120) +- 1, with 5129 digits, by Harvey Dubner.
     * 697053813 * 2^(16352) +- 1, with 4932 digits, found by Indlekofer
       and Ja'rai in 1994.
     * 1691232 * 1001 * 10^(4020) +- 1 with 4030 digits, found by H.
       Dubner.
     * 4650828 * 1001 * 10^(3429) +- 1. Found by H. Dubner as well.
       
   The two largest Sophie Germain primes (i.e. p and 2p+1 are both
   primes) are p = 2687145 * 3003 * 10^(5072) - 1 and q=2p + 1, found by
   Harvey Dubner, in October 3, 1995.
   
      References
      
   B. K. Parady and J. F. Smith and S. E. Zarantonello, Smith, Noll and
   Brown. Largest known twin primes. Mathematics of Computation, vol.55,
   1990, pp. 381-382.
   
Largest Fermat number with known factorization

   F_(11) = (2^(2^(11))) + 1 which was factored by Brent & Morain in
   1988. F_9 = (2^(2^9)) + 1 = 2^(512) + 1 was factored by A.K. Lenstra,
   H.W. Lenstra Jr., M.S. Manasse & J.M. Pollard in 1990. F_(10) was
   factored by Richard Brent who found a 40-digit factor of 2^(1024) + 1
   on October 20, 1995. The cofactor is a 252 digit number, which is not
   so easy to factor. Luckily, this number was also prime.
   
Algorithms to factor integer numbers

   There are several known algorithms that have subexponential estimated
   running time, to mention just a few:
   
     * Continued fraction algorithm.
     * Quadratic sieve algorithm.
     * Class Group method.
     * Elliptic curve algorithm.
     * Number field sieve.
     * Dixon's random squares algorithm.
     * Valle's two-thirds algorithm.
     * Seysen's class group algorithm.
       
      References
      
   A.K. Lenstra, H.W. Lenstra Jr. Algorithms in Number Theory. J. van
   Leeuwen (ed.), Handbook of Theoretical Computer Science, Volume A:
   Algorithms and Complexity Elsevier, pp. 673-715, 1990.
   
Primality Testing

   The problem of primality testing and factorization are two distinct
   problems. If we concentrate on primality testing, we never need to
   know the actual factors. The only question to be answered is "is the
   number in question prime or composite."
   
   Wilson's Theorem: The integer p is prime if and only if (p - 1)! is
   congruent to -1 (mod p)
   
   Since there is no known method for rapidly computing (N - 1)! (mod N)
   in, say, log N steps, so Wilson's characterization of primes is of no
   practical value to the testing of the primality of N.
   
   There are many different primality tests and they can be classified in
   at least three different ways:
   
    1. Tests for numbers of special forms
       versus
       Tests for generic numbers
    2. Tests with full justification
       versus
       Tests with justification based on conjectures
    3. Deterministic tests
       versus
       Probabilistic or Monte Carlo tests
       
   Miller's Test
   
   In 1976, G. L. Miller proposed a primality test, which was justified
   using a generalized form of Riemann's hypothesis.
   
   The APR Test
   
   The primality test devised by L. M. Adleman, C. Pomerance and R. S.
   Rumely (1983), also known as the APR test, represents a breakthrough
   because:
   
    1. It is applicable to arbitrary natural numbers N, without requiring
       the knowledge of factors of N - 1 or N + 1.
    2. The running time t(N) is almost polynomial.
    3. The test is justified rigorously, and for the first time ever in
       this domain, it is necessary to appeal to deep results in the
       theory of algebraic numbers; it involves calculations with roots
       of unity and the general reciprocity law for the power residue
       symbol.
       
   The running time of the APR is at the present the world record for a
   deterministic primality test.
   
   Soon afterwards, H. Cohen & A. K. Lenstra (1984) modified the APR
   test, making it more flexible, using Gauss sums in the proof (instead
   of the reciprocity law), and having the new test programmed for
   practical applications. It was the first primality test in existence
   that can routinely handle numbers of up 100 decimal digits, and it
   does so in about 45 seconds.
   
   Monte Carlo methods
   
   Ribenboim mentions three Monte Carlo tests, due to R. Baillie &
   Wagstaff, Jr. (1980), R. Solovay & V. Strassen (1977), and M. O. Rabin
   (1976, 1980).
   
   Elliptic Curves Primality Proving, ECPP
   
   ECPP stands for "Elliptic Curves and Primality Proving". The algorithm
   is described in:
   
    A. O. L. Atkin and F. Morain
    "Elliptic curves and primality proving"
    To appear.

   It is a deterministic algorithm that gives a certificate of primality
   for numbers that can be as large as 10**1000 (one thousand).
   
   References
   
[1] A. O. L. Atkin and F. Morain
    "Elliptic curves and primality proving"
    To appear in Math. Comp.

%  Lieven Marchand <mal@bewoner.dma.be>

[2] F. Morain
    "Courbes elliptiques et tests de primalite'"
    The`se, Universite' de Lyon I, 1990.
    Available at:
http://lix.polytechnique.fr/~morain/english-index.html

   This subsection is copyright (C) 1995. Harry J. Smith,
   HJSmith@ix.netcom.com.
   
List of record numbers

   Chris Caldwell (caldwell@utm.edu) maintains a list called "The Largest
   Known Primes." Some of the ways to get this list are:
   
    web:     http://www.utm.edu/research/primes/largest.html
    gopher:  unix1.utm.edu, directory 1/user/Public_FTP/pub/math/primes
    ftp:     math.utm.edu, directory /pub/math/primes

   Finger primes@math.utm.edu for a few record primes and the current
   ways to get the lists. He would like to know of any new titanic primes
   (over 1000 digits) so that he can add them to his list.
   
What is the current status on Mersenne primes?

   The following Mersenne primes are known.
   


  Number        p                        Year   Discoverer

     1-4     2,3,5,7                 pre-1500
     5          13                       1461   Anonymous
     6-7        17,19                    1588   Cataldi
     8          31                       1750   Euler
     9          61                       1883   I.M. Pervushin
    10          89                       1911   Powers
    11          107                      1914   Powers
    12          127                      1876   Lucas
    13-14       521,607                  1952   Robinson
    15-17       1279,2203,2281           1952   R. M. Robinson
    18          3217                     1957   Riesel
    19-20       4253,4423                1961   Hurwitz & Selfridge
    21-23       9689,9941,11213          1963   Gillies
    24          19937                    1971   Tuckerman
    25          21701                    1978   Noll & Nickel
    26          23209                    1979   Noll
    27          44497                    1979   Slowinski & Nelson
    28          86243                    1982   Slowinski
    29          110503                   1988   Colquitt & Welsh
    30          132049                   1983   Slowinski
    31          216091                   1985   Slowinski
    32          756839                   1992   Slowinski & Gage
    33          859433                   1994   Slowinski & Gage
    34          1257787                  1996   Slowinski & Gage
    35          1398269                  1996   Armengaud, Woltman, et. al.
    36?         2976221                  1996   Spence, Woltman, et. al.

   The way to determine if 2^p - 1 is prime is to use the Lucas-Lehmer
   test:
   
      Lucas_Lehmer_Test(p):
         u := 4
         for i from 3 to p do
            u := u^2-2 mod 2^p-1
         od
         if u == 0 then
            2^p-1 is prime
         else
            2^p-1 is composite
         fi

   All exponents less than 1,481,800 have now been tested at least once.
   
      References
      
   An introduction to the theory of numbers. G.H. Hardy, E.M. Wright.
   Fifth edition, 1979, Oxford.
   
Formulae to compute prime numbers

   There is no polynomial which gives all the prime numbers. This is a
   simple exercise to prove.
   
   There is no non-constant polynomial that only takes on prime values.
   The proof is simple enough that an high school student could probably
   discover it. See, for example, Ribenboim's book The Book of Prime
   Number Records.
   
   Note, however, by the work of Jones, Sato, Wada, and Wiens, there is a
   polynomial in 26 variables such that the set of primes coincides with
   the set of positive values taken by this polynomial. See Ribenboim,
   pp. 147-150.
   
   But most people would object to the term ``formula" restricted to mean
   polynomial. Can we not use summation signs, factorial, and the floor
   function in our ``formula"? If so, then indeed, there are formulas for
   the prime numbers. Some of them are listed below.
   
   A reasonable interpretation of the word ``formula" is simply ``Turing
   machine that halts on all inputs". Under this interpretation, there
   certainly are halting Turing machines which compute the n-th prime
   number. However, nobody knows how to compute the n-th prime in time
   polynomial in log n. That's still an open question.
   
   Herb Wilf has addressed the question, ``What is a formula?" in his
   article, ``What is an answer?" which appeared in the American
   Mathematical Monthly, 89 (1982), 289-292. He draws a distinction
   between ``formula" and ``good formula". Anyone who claims ``there is
   no formula for the prime numbers" should read this article.
   
   Here are just a few articles that discuss ``formulas" for primes.
   Almost all of these do not require computation of the primes ahead of
   time. Most of them rely on standard mathematical functions such as
   summation, factorial, greatest integer function, etc.
   
      References
      
   C. Isenkrahe. Math. Annalen 53 (1900), 42-44.
   
   W. H. Mills. Bulletin of the American Mathematical Society 53 (1947),
   604.
   
   L. Moser. Mathematics Magazine 23 (1950), 163-164.
   
   E. M. Wright. American Mathematical Monthly 58 (1951), 616-618.
   (Correction, 59 (1952), 99.)
   
   E. M. Wright. Journal of the London Mathematical Society 29 (1954),
   63-71.
   
   B. R. Srinivasan. Journal of the Indian Mathematical Society 25
   (1961), 33-39.
   
   C. P. Willans. Mathematics Gazette 48 (1964), 413-415.
   
   V. C. Harris. Nordisk Mat. Tidskr. 17 (1969), 82.
   
   U. Dudley. American Mathematical Monthly 76 (1969), 23-28.
   
   C. Vanden Eynden. American Mathematical Monthly 79 (1972), 625.
   
   S. W. Golomb. American Mathematical Monthly 81 (1974), 752-754.
   
   Algorithmic Number Theory. J.O. Shallit, E. Bach. (to be published,
   MIT Press).
   
   A Course in Computational Algebraic Number Theory. Henri Cohen.
   Springer-Verlag, Graduate Texts in Math, 1993.


-- 
Alex Lopez-Ortiz                                         alopez-o@unb.ca
http://www.cs.unb.ca/~alopez-o                       Assistant Professor	
Faculty of Computer Science                  University of New Brunswick

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