Prime Numbers

4 Chapter 1 PRIMES!
of the most recent “largest known primes” performed thus: The primality
proof/disproof for a single candidate 2 q − 1 required in 2004 about one CPUweek,
on a typical modern PC (see continually updating website [Woltman
2000]). By contrast, a number of order 2 20000000 would have required, just
a decade earlier, perhaps a decade of a typical PC’s CPU time! Of course,
both machine and algorithm advances are responsible for this performance
offset. To convey again an idea of scale: At the start of the 21st century, a
typical workstation equipped with the right software can multiply together
two numbers, each with a million decimal digits, in a fraction of a second. As
explained at the end of Section 9.5.2, appropriate cluster hardware can now
multiply two numbers each of a billion digits in roughly one minute.
The special Mersenne form 2 q − 1 of such numbers renders primality
proofs feasible. For Mersenne numbers we have the very speedy Lucas–
Lehmer test, discussed in Chapter 4. What about primes of no special form—
shall we say “random” primes? Primality proofs can be effected these days
for such primes having a few thousand digits. Much of the implementation
work has been pioneered by F. Morain, who applied ideas of A. Atkin
and others to develop an efficient elliptic curve primality proving (ECPP)
method, along with a newer “fastECPP” method, discussed in Chapter 7. A
typically impressive ECPP result at the turn of the century was the proof
that (2 7331 − 1)/458072843161, possessed of 2196 decimal digits, is prime (by
Mayer and Morain; see [Morain 1998]). A sensational announcement in July
2004 by Franke, Kleinjung, Morain, and Wirth is that, thanks to fastECPP,
the Leyland number
4405 2638 + 2638 4405 ,
having 15071 decimal digits, is now proven prime.
Alongside these modern factoring achievements and prime-number analyses
there stand a great many record-breaking attempts geared to yet more
specialized cases. From time to time we see new largest twin primes (pairs of
primes p, p+2), an especially long arithmetic progression {p, p+d,...,p+kd}
of primes, or spectacular cases of primes falling in other particular patterns.
There are searches for primes we expect some day to find but have not yet
found (such as new instances of the so-called Wieferich, Wilson, or Wall–Sun–
Sun primes). In various sections of this book we refer to a few of these many
endeavors, especially when the computational issues at hand lie within the
scope of the book.
Details and special cases aside, the reader should be aware that there
is a widespread “culture” of computational research. For a readable and
entertaining account of prime number and factoring “records,” see, for
example, [Ribenboim 1996] as well as the popular and thorough newsletter
of S. Wagstaff, Jr., on state-of-the-art factorizations of Cunningham numbers
(numbers of the form b n ± 1forb ≤ 12). A summary of this newsletter is
kept at the website [Wagstaff 2004]. Some new factorization records as of this
(early 2005) writing are the following: