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From: [email protected] (Alex Lopez-Ortiz)
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Subject: 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 <[email protected]>

[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,
[email protected].

List of record numbers

Chris Caldwell ([email protected]) 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 [email protected] 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 [email protected]
http://www.cs.unb.ca/~alopez-o Assistant Professor
Faculty of Computer Science University of New Brunswick