The set of all prime numbers.
Prime.each(100) do |prime| p prime #=> 2, 3, 5, 7, 11, ...., 97 end
Prime
is Enumerable:
Prime.first 5 # => [2, 3, 5, 7, 11]
For convenience, each instance method of Prime
.instance can be accessed as a class method of Prime
.
e.g.
Prime.instance.prime?(2) #=> true Prime.prime?(2) #=> true
A “generator” provides an implementation of enumerating pseudo-prime numbers and it remembers the position of enumeration and upper bound. Furthermore, it is an external iterator of prime enumeration which is compatible with an Enumerator
.
Prime
::PseudoPrimeGenerator
is the base class for generators. There are few implementations of generator.
Prime
::EratosthenesGenerator
Uses eratosthenes' sieve.
Prime
::TrialDivisionGenerator
Uses the trial division method.
Prime
::Generator23
Generates all positive integers which are not divisible by either 2 or 3. This sequence is very bad as a pseudo-prime sequence. But this is faster and uses much less memory than the other generators. So, it is suitable for factorizing an integer which is not large but has many prime factors. e.g. for Prime#prime?
.
Iterates the given block over all prime numbers.
ubound
Optional. An arbitrary positive number. The upper bound of enumeration. The method enumerates prime numbers infinitely if ubound
is nil.
generator
Optional. An implementation of pseudo-prime generator.
An evaluated value of the given block at the last time. Or an enumerator which is compatible to an Enumerator
if no block given.
Calls block
once for each prime number, passing the prime as a parameter.
ubound
Upper bound of prime numbers. The iterator stops after it yields all prime numbers p <= ubound
.
Re-composes a prime factorization and returns the product.
pd
Array
of pairs of integers. The each internal pair consists of a prime number – a prime factor – and a natural number – an exponent.
For [[p_1, e_1], [p_2, e_2], ...., [p_n, e_n]]
, it returns:
p_1**e_1 * p_2**e_2 * .... * p_n**e_n. Prime.int_from_prime_division([[2,2], [3,1]]) #=> 12
Returns true if value
is a prime number, else returns false.
value
an arbitrary integer to be checked.
generator
optional. A pseudo-prime generator.
Returns the factorization of value
.
value
An arbitrary integer.
generator
Optional. A pseudo-prime generator. generator
.succ must return the next pseudo-prime number in the ascending order. It must generate all prime numbers, but may also generate non prime numbers too.
ZeroDivisionError
when value
is zero.
For an arbitrary integer:
n = p_1**e_1 * p_2**e_2 * .... * p_n**e_n,
prime_division
(n) returns:
[[p_1, e_1], [p_2, e_2], ...., [p_n, e_n]]. Prime.prime_division(12) #=> [[2,2], [3,1]]