Prime Numbers

118 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
3.1.2 Trial division
Trial division is the method of sequentially trying test divisors into a number
n so as to partially or completely factor n. We start with the first prime, the
number 2, and keep dividing n by 2 until it does not go, and then we try the
next prime, 3, on the remaining unfactored portion, and so on. If we reach a
trial divisor that is greater than the square root of the unfactored portion, we
may stop, since the unfactored portion is prime.
Here is an example. We are given the number n = 7399. We trial divide
by 2, 3, and 5 and find that they are not factors. The next choice is 7. It
is a factor; the quotient is 1057. We next try 7 again, and find that again it
goes, the quotient being 151. We try 7 one more time, but it is not a factor
of 151. The next trial is 11, and it is not a factor. The next trial is 13, but
this exceeds the square root of 151, so we find that 151 is prime. The prime
factorization of 7399 is 7 2 · 151.
It is not necessary that the trial divisors all be primes, for if a composite
trial divisor d is attempted, where all the prime factors of d have previously
been factored out of n, then it will simply be the case that d is not a factor
when it is tried. So though we waste a little time, we are not led astray in
finding the prime factorization.
Let us consider the example n = 492. We trial divide by 2 and find that
it is a divisor, the quotient being 246. We divide by 2 again and find that
the quotient is 123. We divide by 2 and find that it does not go. We divide
by 3, getting the quotient 41. We divide by 3, 4, 5 and 6 and find they do
not go. The next trial is 7, which is greater than √ 41, so we have the prime
factorization 492 = 2 2 · 3 · 41.
Now let us consider the neighboring number n = 491. We trial divide by
2, 3, and so on up through 22 and find that none are divisors. The next trial
is 23, and 23 2 > 491, so we have shown that 491 is prime.
To speed things up somewhat, one may exploit the fact that after 2, the
primes are odd. So 2 and the odd numbers may be used as trial divisors. With
n = 491, such a procedure would have stopped us from trial dividing by the
even numbers from 4 to 22. Here is a short description of trial division by 2
andtheoddintegersgreaterthan2.
Algorithm 3.1.1 (Trial division). We are given an integer n > 1. This
algorithm produces the multiset F of the primes that divide n. (A “multiset”
is a set where elements may be repeated; that is, a set with multiplicities.)
1. [Divide by 2]
F = {}; // The empty multiset.
N = n;
while(2|N) {
N = N/2;
F = F∪{2};
}
2. [Main division loop]