Prime Numbers

1.1 Problems and progress 5
The Pollard-(p − 1) method (see our Section 5.4) was used in 2003 by
P. Zimmermann to find 57-digit factors of two separate numbers, namely
6 396 + 1 and 11 260 +1.
There are recent successes for the elliptic-curve method (ECM) (see our
Section 7.4.1), namely, a 57-digit factor of 2 997 − 1 (see [Wagstaff 2004]), a
58-digit factor of 8 · 10 141 − 1 found in 2003 by R. Backstrom, and a 59digit
factor of 10 233 − 1 found in 2005 by B. Dodson. (It is surprising, and
historically rare over the last decade, that the (p − 1) method be anywhere
near the ECM in the size of record factors.)
In late 2001, the quadratic sieve (QS) (see our Section 6.1), actually a threelarge-prime
variant, factored a 135-digit composite piece of 2 803 − 2 402 +1.
This seems to have been in some sense a “last gasp” for QS, being as the
more modern NFS and SNFS have dominated for numbers of this size.
The general-purpose number field sieve (GNFS) has, as we mentioned earlier,
factored the 174-digit number RSA-576. For numbers of special form, the
special number field sieve (SNFS) (see our Section 6.2.7) has factored
numbers beyond 200 digits, the record currently being the 248-digit number
2 821 +2 411 +1.
Details in regard to some such record factorizations can be found in the
aforementioned Wagstaff newsletter. Elsewhere in the present book, for
example after Algorithm 7.4.4 and at other similar junctures, one finds older
records from our 1st edition; we have left these intact because of their historical
importance. After all, one wants not only to see progress, but also track it.
Here at the dawn of the 21st century, vast distributed computations are
not uncommon. A good lay reference is [Peterson 2000]. Another lay treatment
about large-number achievements is [Crandall 1997a]. In the latter exposition
appears an estimate that answers roughly the question, “How many computing
operations have been performed by all machines across all of world history?”
One is speaking of fundamental operations such as logical “and” as well as
“add,” “multiply,” and so on. The answer is relevant for various issues raised
in the present book, and could be called the “mole rule.” To put it roughly,
right around the turn of the century (2000 ad), about one mole—that is, the
Avogadro number 6 · 10 23 of chemistry, call it 10 24 —is the total operation
count for all machines for all of history. In spite of the usual mystery and
awe that surrounds the notion of industrial and government supercomputing,
it is the huge collection of personal computers that allows this 10 24 ,this
mole. The relevance is that a task such as trial dividing an integer N ≈ 10 50
directly for prime factors is hopeless in the sense that one would essentially
have to replicate the machine effort of all time. To convey an idea of scale,
a typical instance of the deepest factoring or primality-proving runs of the
modern era involves perhaps 10 16 to 10 18 machine operations. Similarly, a fulllength
graphically rendered synthetic movie of today—for example, the 2003
Pixar/Disney movie Finding Nemo—involves operation counts in the 10 18
range. It is amusing that for this kind of Herculean machine effort one may
either obtain a single answer (a factor, maybe even a single “prime/composite”