Integral values fall into two categories: prime numbers, whose only integral divisors are their own values and 1, and composite numbers, which also have at least one other (prime number) integral divisor. All composite integral values can be factorized as the product of prime numbers. E.g., 6 can be factorized as 2 * 3; 8 can be factorized as 2 * 2 * 2. Finding these prime factors is called the prime number factorization, or `prime factorization'. When factorizing a value its prime factors may sometimes repeatedly be used as integral divisors: 8 is factorized as pow(2, 3), and 36 is factorized as
36 = pow(2, 2) * pow(3, 2)
The class FBB::PrimeFactors performs prime number factorizations of FBB::BigInt values. When factorizing a value prime numbers up to sqrt(value) must be available, as prime numbers up to sqrt(value) may be factors of value. Currently PrimeFactors uses the sieve of Eratosthenes to find these prime numbers. To find the next prime number beyond lastPrime, the sieve of Eratosthenes must be used repeatedly for lastPrime += 2 until lastPrime is prime. Once determined, prime numbers can of course be used directly to determine the next prime number or to factorize an integral value. To accellerate prime number factorization and Eratosthenes's sieve PrimeFactors saves all its computed prime numbers in either a std::vector or in a file. Once determined, these prime numbers may again be used when factorizing the next integral value.
After factorizing an integral value its prime number factors and associated powers are made available in a vector of (PrimeFactors::PrimePower) structs, containing the value's sorted prime factors and associated powers.
struct PrimePower
{
BigInt prime;
size_t power;
};
Here, power represents the number of times prime can be used
as an integral divisor of the value that was factorized by PrimeFactors.
Initially the BigIntVector passed as argument may be empty or may contain at least two primes (which must be, respectively, 2 and 3). The prime numbers in primes must be sorted. The constructor does not verify whether the prime numbers are actually sorted, but if the BigIntVector contains primes it does check whether the first two prime numbers are indeed 2 and 3. An FBB::Exception is thrown if this condition is not met.
While numbers are being factorized, new prime numbers may be added to primes, and primes can be reused by othher PrimeFactors objects.
Primes are read from the named stream in blocks of at most blockSize, and new primes are flushed to this stream once blockSize new primes have been generated or when the PrimeFactors object (i.e., the last PrimeFactors object sharing the stream) ceases to exist.
If the stream does not yet exist it is created by PrimeFactors. The stream may either be empty, or it must contain sorted and white-space delimited prime numbers (inserted as hexadecimal BigInt values). The first two primes on this file must be, respectively, 2 and 3. The constructor does not verify whether the prime numbers are actually sorted, but if the stream contains primes it does check whether the first two prime numbers are indeed 2 and 3. An FBB::Exception is thrown if this condition is not met.
While numbers are being factorized, new prime numbers may be added to the stream, and the stream can be reused by other PrimeFactors objects.
The default copy and move constructors are available. FBB::PrimeFactor objects created using the copy constructor share the prime numbers storage device (the BigIntVector or the stream containing the primes) with their source objects.
The default copy and move assignment operators are available. Left-hand side FBB::PrimeFactor objects receiving values from right-hand side FBB::PrimeFactor objects using the copy assignment operator share the prime numbers storage device (the BigIntVector or the stream containing the primes) with their right-hand side FBB::PrimeFactors arguments.
#include <iostream>
#include <bobcat/primefactors>
using namespace std;
using namespace FBB;
int main(int argc, char **argv)
{
PrimeFactors pf1("/tmp/primes");
PrimeFactors::Factors const *factors = &pf1.factorize(stoull(argv[1]));
cout << "Using /tmp/primes:\n";
for (auto &factor: *factors)
cout << factor.prime << "**" << factor.power << ' ';
vector<BigInt> primes;
PrimeFactors pf2(primes);
factors = &pf2.factorize(stoull(argv[1]));
cout << "\n"
"Using BigIntVector:\n";
for (auto &factor: *factors)
cout << factor.prime << "**" << factor.power << ' ';
cout << "\n"
"Collected primes: ";
for (auto &prime: primes)
cout << prime << ' ';
cout << '\n';
}
If this program is run with argument 1950 it produces the following output:
Using /tmp/primes:
2**1 3**1 5**2 13**1
Using BigIntVector:
2**1 3**1 5**2 13**1
Collected primes: 2 3 5 7 11 13