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# 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.

##### 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.
• 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

``` A. O. L. Atkin and F. Morain
"Elliptic curves and primality proving"
To appear in Math. Comp.

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

 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.    Next: Special Numbers and Functions Up: Number Theory Previous: Fermat's Last Theorem

Alex Lopez-Ortiz
Fri Feb 20 21:45:30 EST 1998