Prime Numbers

3.2 Sieving 123
this number’s entry is also 3. Dividing by 3 again, we get 1337, whose entry
is 7. Dividing by 7, we get 191, whose entry is 1. Thus 191 is prime and we
have the prime factorization
12033 = 3 2 · 7 · 191.
Factor tables predate, by far, the computer era. Extensive hand-computed
factor tables were indispensable to researchers doing numerical work in
number theory for many decades prior to the advent of electronic calculating
engines.
3.2.4 Sieving to construct complete factorizations
Again, at the cost of somewhat more space, but very little more time, one
may adapt the sieve of Eratosthenes so that next to entry m is the complete
prime factorization of m. One does this by appending the prime p to lists at
locations p, 2p, 3p,..., p⌊N/p⌋. One also needs to sieve with the powers pa of
primes p ≤ √ N,wherethepowerpadoes not exceed N. At each multiple of
pa another copy of the prime p is appended. To avoid sieving with the primes
√N,N
in the interval , one can divide to complete the factorization.
For example, say N = 20000; let us follow what happens to the entry
m = 12033. Sieving by 3, we change the 1 at location 12033 to 3. Sieving by
9, we change the 3 at location 12033 to 3, 3. Sieving by 7, we change the entry
to 3, 3, 7. At the end of sieving (which includes sieving with all primes up to
139 and higher powers of these primes up to 20000), we return to each location
in the sieve and multiply the list there. At the location 12033, we multiply
3 · 3 · 7, getting 63. Dividing 63 into 12033, the quotient is 191, which is also
put on the list. So the final list for 12033 is 3, 3, 7, 191, giving the complete
prime factorization of 12033.
3.2.5 Sieving to recognize smooth numbers
Using the sieve of Eratosthenes to get complete factorizations may be
simplified and turned into a device to recognize all of the B-smooth numbers
(see Definition 1.4.8) in [2,N]. We suppose that 2 ≤ B ≤ √ N. Perform the
factorization sieve as in the above subsection, but with two simplifications: (1)
Do not sieve with any p a where p exceeds B, and (2) if the product of the list
at a location is not equal to that location number, then do not bother dividing
to get the quotient. The B-smooth numbers are precisely those at locations
that are equal to the product of the primes in the list at that location.
To simplify slightly, we might multiply a running product at each location
by p whenever p a hits there. There is no need to keep the lists around if we
are interested only in picking out the B-smooth numbers. At the end of the
sieve, those locations whose location numbers are equal to the entry in the
location are the B-smooth numbers.
For example, say B =10andN = 20000. The entry corresponding to
12033 starts as 1, and gets changed sequentially to 3, to 9, and finally to