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mersenne.txt
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1995-10-11
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3Q: What are the values of:
largest known Mersenne prime?
A: It is 2^859433-1. It was discovered by a Cray in 1994.
It has 258,716 digits.
1 2^859433-1 258716 SG 94 Mersenne ?
What is the current status on Mersenne primes?
A: 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.
nr p year by
-----------------------------------------------------------------
1-5 2,3,5,7,13 in or before the middle ages
6-7 17,19 1588 Cataldi
8 31 1750 Euler
9 61 1883 Pervouchine
10 89 1911 Powers
11 107 1914 Powers
12 127 1876 Lucas
13-14 521,607 1952 Robinson
15-17 1279,2203,2281 1952 Lehmer
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 jr.
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 and 430K - 520K
More on Mersenne primes and the Lucas-Lehmer test can be found in:
G.H. Hardy, E.M. Wright, An introduction to the theory of numbers,
fifth edition, 1979, pp. 16, 223-225.
(Please send updates to alopez-o@maytag.UWaterloo.ca)
Alex Lopez-Ortiz alopez-o@maytag.UWaterloo.ca
Department of Computer Science University of Waterloo
Waterloo, Ontario Canada