Prime Numbers

1.3 Primes of special form 23
for various small primes r ≡ 1(modq)andr ≡±1 (mod 8). For the survivors,
one then invokes the celebrated Lucas–Lehmer test, which is a rigorous
primality test (see Section 4.2.1).
As of this writing, the known Mersenne primes are those displayed in
Table 1.2.
2 2 − 1 2 3 − 1 2 5 − 1 2 7 − 1
2 13 − 1 2 17 − 1 2 19 − 1 2 31 − 1
2 61 − 1 2 89 − 1 2 107 − 1 2 127 − 1
2 521 − 1 2 607 − 1 2 1279 − 1 2 2203 − 1
2 2281 − 1 2 3217 − 1 2 4253 − 1 2 4423 − 1
2 9869 − 1 2 9941 − 1 2 11213 − 1 2 19937 − 1
2 21701 − 1 2 23209 − 1 2 44497 − 1 2 86243 − 1
2 110503 − 1 2 132049 − 1 2 216091 − 1 2 756839 − 1
2 859433 − 1 2 1257787 − 1 2 1398269 − 1 2 2976221 − 1
2 3021377 − 1 2 6972593 − 1 2 13466917 − 1 2 20996011 − 1
2 24036583 − 1 2 25964951 − 1
Table 1.2 Known Mersenne primes (as of Apr 2005), ranging in size from 1 decimal
digit to over 7 million decimal digits.
Over the years 1979–96, D. Slowinski found seven Mersenne primes, all
of the Mersenne primes from 2 44497 − 1 to 2 1257787 − 1, inclusive, except
for 2 110503 − 1 (the first of the seven was found jointly with H. Nelson and
the last three with P. Gage). The “missing” prime 2 110503 − 1 was found by
W. Colquitt and L. Welsh, Jr., in 1988. The record for consecutive Mersenne
primes is still held by R. Robinson, who found the five starting with 2 521 − 1
in 1952. The prime 2 1398269 − 1 was found in 1996 by J. Armengaud and
G. Woltman, while 2 2976221 −1 was found in 1997 by G. Spence and Woltman.
The prime 2 3021377 − 1 was discovered in 1998 by R. Clarkson, Woltman,
S. Kurowski, et al. (further verified by D. Slowinski as prime in a separate
machine/program run). Then in 1999 the prime 2 6972593 − 1 was found by
N. Hajratwala, Woltman, and Kurowski, then verified by E. Mayer and
D. Willmore. The case 2 13466917 − 1 was discovered in November 2001 by
M. Cameron, Woltman, and Kurowski, then verified by Mayer, P. Novarese,
and G. Valor. In November 2003, M. Shafer, Woltman, and Kurowski found
2 20996011 − 1. The Mersenne prime 2 24036583 − 1 was found in May 2004 by
J. Findley, Woltman, and Kurowski. Then in Feb 2005, M. Nowak, Woltman
and Kurowski found 2 25964951 −1. Each of these last two Mersenne primes has
more than 7 million decimal digits.
The eight largest known Mersenne primes were found using a fast
multiplication method—the IBDWT—discussed in Chapter 8.8 (Theorem