Prime Numbers

124 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
63. Thus 12033 is not 10-smooth. However, the entry at 12000 gets changed
sequentially to 2, 4, 8, 16, 32, 96, 480, 2400, and finally 12000. Thus 12000 is
10-smooth.
One important way of speeding this sieve is to do the arithmetic at each
location in the sieve with logarithms. Doing exact arithmetic with logarithms
involves infinite precision, but there is no need to be exact. For example, say we
use the closest integer to the base-2 logarithm. For 12000 this is 14. We also use
the approximations lg 2 ≈ 1 (this one being exact), lg 3 ≈ 2, lg 5 ≈ 2, lg 7 ≈ 3.
The entry now at location 12000 gets changed sequentially to 1, 2, 3, 4, 5, 7,
9, 11, 13. This is close enough to the target 14 for us to recognize that 12000 is
smooth. In general, if we are searching for B-smooth numbers, then an error
smaller than lg B is of no consequence.
One should see the great advantage of working with approximate
logarithms, as above. First, the numbers one deals with are very small. Second,
the arithmetic necessary is addition, an operation that is much faster to
perform on most computers than multiplication or division. Also note that
the logarithm function moves very slowly for large arguments, so that all
nearby locations in the sieve have essentially the same target. For example,
above we had 14 the target for 12000. This same number is used as the target
for all locations between 2 13.5 and 2 14.5 , namely, all integers between 11586
and 23170.
We shall find later an important application for this kind of sieve in
factorization algorithms. And, as discussed in Section 6.4, sieving for smooth
numbers is also crucial in some discrete logarithm algorithms. In these settings
we are not so concerned with doing a perfect job sieving, but rather just
recognizing most B-smooth numbers without falsely reporting too many
numbers that are not B-smooth. This is a liberating thought that allows
further speed-ups in sieving. The time spent sieving with a prime p in the
sieve is proportional to the product of the length of the sieve and 1/p. In
particular, small primes are the most time-consuming. But their logarithms
contribute very little to the sum, and so one might agree to forgo sieving
with these small primes, allowing a little more error in the sieve. In the above
example, say we forgo sieving with the moduli 2, 3, 4, 5. We will sieve by
higher powers of 2, 3, and 5, as well as all powers of 7, to recognize our 10smooth
numbers. Then the running sum in location 12000 is 3, 4, 5, 9, 11.
This total is close enough to 14 to cause a report, and the number 12000 is
not overlooked. But we were able to avoid the most costly part of the sieve to
find it.
3.2.6 Sieving a polynomial
Suppose f(x) is a polynomial with integer coefficients. Consider the numbers
f(1),f(2),...,f(N). Say we wish to find the prime numbers in this list, or to
prepare a factor table for the list, or to find the B-smooth numbers in this
list. All of these tasks can easily be accomplished with a sieve. In fact, we have
already seen a special case of this for the polynomial f(x) =2x +1, when we