Prime Numbers

3.1 Trial division 119
d =3;
while(d 2 ≤ N) {
while(d|N) {
N = N/d;
F = F∪{d};
}
d = d +2;
}
if(N == 1) return F;
return F∪{N};
After 3, primes are either 1 or 5 (mod 6), and one may step through the
sequence of numbers that are 1 or 5 (mod 6) by alternately adding 2 and 4 to
the latest number. This is a special case of a “wheel,” which is a finite sequence
of addition instructions that may be repeated indefinitely. For example, after
5, all primes may be found in one of 8 residue classes (mod 30), and a wheel
that traverses these classes (beginning from 7) is
4, 2, 4, 2, 4, 6, 2, 6.
Wheels grow more complicated at a rapid rate. For example, to have a wheel
that traverses the numbers that are coprime to all the primes below 30, one
needs to have a sequence of 1021870080 numbers. And in comparison with
thesimple2, 4 wheel based on just the two primes 2 and 3, we save only little
more than 50% of the trial divisions. (Specifically, about 52.6% of all numbers
coprime to 2 and 3 have a prime factor less than 30.) It is a bit ridiculous
to use such an ungainly wheel. If one is concerned with wasting time because
of trial division by composites, it is much easier and more efficient to first
prepare a list of the primes that one will be using for the trial division. In the
next section we shall see efficient ways to prepare this list.
3.1.3 Practical considerations
It is perfectly reasonable to use trial division as a primality test when n is
not too large. Of course, “too large” is a subjective quality; such judgment
depends on the speed of the computing equipment and how much time you are
willing to allow a computer to run. It also makes a difference whether there is
just the occasional number you are interested in, as opposed to the possibility
of calling trial division repeatedly as a subroutine in another algorithm. On a
modern workstation, and very roughly speaking, numbers that can be proved
prime via trial division in one minute do not exceed 13 decimal digits. In one
day of current workstation time, perhaps a 19-digit number can be resolved.
(Although these sorts of rules of thumb scale, naturally, according to machine
performance in any given era.)
Trial division may also be used as an efficient means of obtaining a partial
factorization n = FR as discussed above. In fact, for every fixed trial division
bound B ≥ 2, at least one quarter of all numbers have a divisor F that is