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-3.0.5/lib/prime.rb, line 212
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
ubound
is 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-3.0.5/lib/prime.rb, line 220
def include?(obj)
case obj
when Integer
prime?(obj)
when Module
Module.instance_method(:include?).bind(Prime).call(obj)
else
false
end
end
# File tmp/rubies/ruby-3.0.5/lib/prime.rb, line 268
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.
For the decomposition:
[[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.
Parameters
pd
-
Array
of pairs of integers. Each pair consists of a prime number – a prime factor – and a natural number – its exponent (multiplicity).
Example
Prime.int_from_prime_division([[3, 2], [5, 1]]) #=> 45 3**2 * 5 #=> 45
# File tmp/rubies/ruby-3.0.5/lib/prime.rb, line 238
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. Integer#prime?
is much more performant.
Parameters
value
-
an arbitrary integer to be checked.
generator
-
optional. A pseudo-prime generator.
# File tmp/rubies/ruby-3.0.5/lib/prime.rb, line 303
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
.
For an arbitrary integer:
p_1**e_1 * p_2**e_2 * ... * p_n**e_n,
prime_division
returns an array of pairs of integers:
[[p_1, e_1], [p_2, e_2], ..., [p_n, e_n]].
Each pair consists of a prime number – a prime factor – and a natural number – its exponent (multiplicity).
Parameters
value
-
An arbitrary integer.
generator
-
Optional. A pseudo-prime generator.
generator
.succ must return the next pseudo-prime number in ascending order. It must generate all prime numbers, but may also generate non-prime numbers, too.
Exceptions
ZeroDivisionError
-
when
value
is zero.
Example
Prime.prime_division(45) #=> [[3, 2], [5, 1]] 3**2 * 5 #=> 45