Subject: sci.math FAQ: Prime Numbers 
Summary: Part 9 of many, New version,
Date: Fri, 17 Nov 1995 17:14:20 GMT
Nntp-Posting-Host: neumann.uwaterloo.ca

Archive-Name: sci-math-faq/primes
Version: 6.2



   
   
   
   
                                PRIME NUMBERS
                                       
   
  Contents:
 
   
     _________________________________________________________________
   
     * Largest known Mersenne prime
     * Largest known prime
     * Largest known twin primes
     * Largest Fermat number with known factorization
     * Algorithms to factor integer numbers
     * List of record numbers
     * What is the current status on Mersenne primes?
     * Formulae to compute prime numbers
       
   
     _________________________________________________________________
   
  
  

Largest known Mersenne prime



   2^(859433) - 1 is prime. It was discovered in 1994.


     _________________________________________________________________



 
Largest known prime



   The largest known prime is the Mersenne prime described above. The
   largest known non-Mersenne prime, is 391581*2^(216193) - 1 .
   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.



   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 largest known twin primes are 697053813 * 2^(16352) +- 1 , with
   4932 digits, found by Indlekofer and Ja'rai in 1994.

   Recent record holders are:

     * 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 = 157324389 * 2^(16352) - 1 and p = 470943129 *
   2^(16352) - 1 , with 4932 digits each, also found by Indlekofer and
   Jarai, in 1994-1995.



   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. The factorization for
   F_(10) is not known.


     _________________________________________________________________


Algorithms to factor integer numbers



   There are several known algorithms that have subexponential estimated
   running time, to mention just a few:

     * Continued fraction algorithm.
     * Class group method.
     * 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.




     _________________________________________________________________


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/departments/math/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?



   Mersenne primes are primes of the form 2^p - 1 . For 2^p - 1 to be
   prime we must have that p is prime. 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





   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



   The following ranges have been checked completely: 2 - 355K,
   360K-386K, and 430K - 520K.



   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.






     _________________________________________________________________



   
   


