The set of all prime numbers.
Example
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]
Retrieving the instance
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
Generators
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?.
# File tmp/rubies/ruby-2.6.10/lib/prime.rb, line 139
def each(ubound = nil, generator = EratosthenesGenerator.new, &block)
generator.upper_bound = ubound
generator.each(&block)
end
Iterates the given block over all prime numbers.
Parameters
ubound-
Optional. An arbitrary positive number. The upper bound of enumeration. The method enumerates prime numbers infinitely if
uboundis nil. generator-
Optional. An implementation of pseudo-prime generator.
Return value
An evaluated value of the given block at the last time. Or an enumerator which is compatible to an Enumerator if no block given.
Description
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.
# File tmp/rubies/ruby-2.6.10/lib/prime.rb, line 175
def int_from_prime_division(pd)
pd.inject(1){|value, (prime, index)|
value * prime**index
}
end
Re-composes a prime factorization and returns the product.
Parameters
pd-
Arrayof pairs of integers. The each internal pair consists of a prime number – a prime factor – and a natural number – an exponent.
Example
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
# File tmp/rubies/ruby-2.6.10/lib/prime.rb, line 151
def prime?(value, generator = Prime::Generator23.new)
raise ArgumentError, "Expected a prime generator, got #{generator}" unless generator.respond_to? :each
raise ArgumentError, "Expected an integer, got #{value}" unless value.respond_to?(:integer?) && value.integer?
return false if value < 2
generator.each do |num|
q,r = value.divmod num
return true if q < num
return false if r == 0
end
end
Returns true if value is a prime number, else returns false.
Parameters
value-
an arbitrary integer to be checked.
generator-
optional. A pseudo-prime generator.
# File tmp/rubies/ruby-2.6.10/lib/prime.rb, line 205
def prime_division(value, generator = Prime::Generator23.new)
raise ZeroDivisionError if value == 0
if value < 0
value = -value
pv = [[-1, 1]]
else
pv = []
end
generator.each do |prime|
count = 0
while (value1, mod = value.divmod(prime)
mod) == 0
value = value1
count += 1
end
if count != 0
pv.push [prime, count]
end
break if value1 <= prime
end
if value > 1
pv.push [value, 1]
end
pv
end
Returns the factorization of value.
Parameters
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.
Exceptions
ZeroDivisionError-
when
valueis zero.
Example
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]]