Prime Numbers

120 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
greater than B and composed solely of primes not exceeding B; see Exercise
3.4.
Trial division is a simple and effective way to recognize smooth numbers,
or numbers without large prime factors, see Definition 1.4.8.
It is sometimes useful to have a “smoothness test,” where for some
parameter B, one wishes to know whether a given number n is B-smooth,
that is, n has no prime factor exceeding B. Trial division up to B not only
tells us whether n is B-smooth, it also provides us with the prime factorization
of the largest B-smooth divisor of n.
The emphasis in this chapter is on recognizing primes and composites,
and not on factoring. So we leave a further discussion of smoothness tests to
a later time.
3.1.4 Theoretical considerations
Suppose we wish to use trial division to completely factor a number n into
primes. What is the worst case running time? This is easy, for the worst case is
when n is prime and we must try as potential divisors the numbers up to √ n.
If we are using just primes as trial divisors, the number of divisions is about
2 √ n/ ln n. If we use 2 and the odd numbers as trial divisors, the number of
divisions is about 1√
1
2 n. If we use a wheel as discussed above, the constant 2
is replaced by a smaller constant.
So this is the running time for trial division as a primality test. What is its
complexity as an algorithm to obtain the complete factorization of n when n is
composite? The worst case is still about √ n, for just consider the numbers that
are the double of a prime. We can also ask for the average case complexity for
factoring composites. Again, it is almost √ n, since the average is dominated
by those composites that have a very large prime factor. But such numbers are
rare. It may be interesting to throw out the 50% worst numbers and compute
the average running time for trial division to completely factor the remaining
numbers. This turns out to be nc ,wherec =1/(2 √ e) ≈ 0.30327; see Exercise
3.5.
As we shall see later in this chapter and in the next chapter, the problem
of recognizing primes is much easier than the general case of factorization. In
particular, we have much better ways than trial division to recognize primes.
Thus, if one uses trial division as a factorization method, one should augment
it with a faster primality test whenever a new unfactored portion of n is
discovered, so that the last bit of trial division may be skipped when the
last part turns out to be prime. So augmenting trial division, the time to
completely factor a composite n essentially is the square root of the second
largest prime factor of n.
Again the average is dominated by a sparse set of numbers, in this case
those numbers that are the product of two primes of the same order of
magnitude; the average being about √ n. But now throwing out the 50% worst
numbers gives a smaller estimate for the average of the remaining numbers.
It is nc ,wherec≈0.23044.