# class Prime

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? .

### Public Instance Methods

each(ubound = nil, generator = EratosthenesGenerator.new, &block) click to toggle source

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 lib/prime.rb, line 136
def each(ubound = nil, generator = EratosthenesGenerator.new, &block)
generator.upper_bound = ubound
generator.each(&block)
end```
int_from_prime_division(pd) click to toggle source

Re-composes a prime factorization and returns the product.

## Parameters¶ ↑

`pd`

Array of 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 lib/prime.rb, line 172
def int_from_prime_division(pd)
pd.inject(1){|value, (prime, index)|
value * prime**index
}
end```
prime?(value, generator = Prime::Generator23.new) click to toggle source

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 lib/prime.rb, line 148
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```
prime_division(value, generator = Prime::Generator23.new) click to toggle source

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 `value` is 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]]```
```# File lib/prime.rb, line 202
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

# An abstract class for enumerating pseudo-prime numbers.
#
# Concrete subclasses should override succ, next, rewind.
class PseudoPrimeGenerator
include Enumerable

def initialize(ubound = nil)
@ubound = ubound
end

def upper_bound=(ubound)
@ubound = ubound
end
def upper_bound
@ubound
end

# returns the next pseudo-prime number, and move the internal
# position forward.
#
# +PseudoPrimeGenerator+#succ raises +NotImplementedError+.
def succ
raise NotImplementedError, "need to define `succ'"
end

# alias of +succ+.
def next
raise NotImplementedError, "need to define `next'"
end

# Rewinds the internal position for enumeration.
#
# See +Enumerator+#rewind.
def rewind
raise NotImplementedError, "need to define `rewind'"
end

# Iterates the given block for each prime number.
def each
return self.dup unless block_given?
if @ubound
last_value = nil
loop do
prime = succ
break last_value if prime > @ubound
last_value = yield prime
end
else
loop do
yield succ
end
end
end

# see +Enumerator+#with_index.
def with_index(offset = 0)
return enum_for(:with_index, offset) { Float::INFINITY } unless block_given?
return each_with_index(&proc) if offset == 0

each do |prime|
yield prime, offset
offset += 1
end
end

# see +Enumerator+#with_object.
def with_object(obj)
return enum_for(:with_object, obj) { Float::INFINITY } unless block_given?
each do |prime|
yield prime, obj
end
end

def size
Float::INFINITY
end
end

# An implementation of +PseudoPrimeGenerator+.
#
# Uses +EratosthenesSieve+.
class EratosthenesGenerator < PseudoPrimeGenerator
def initialize
@last_prime_index = -1
super
end

def succ
@last_prime_index += 1
EratosthenesSieve.instance.get_nth_prime(@last_prime_index)
end
def rewind
initialize
end
alias next succ
end

# An implementation of +PseudoPrimeGenerator+ which uses
# a prime table generated by trial division.
class TrialDivisionGenerator < PseudoPrimeGenerator
def initialize
@index = -1
super
end

def succ
TrialDivision.instance[@index += 1]
end
def rewind
initialize
end
alias next succ
end

# Generates all integers which are greater than 2 and
# are not divisible by either 2 or 3.
#
# This is a pseudo-prime generator, suitable on
# checking primality of an integer by brute force
# method.
class Generator23 < PseudoPrimeGenerator
def initialize
@prime = 1
@step = nil
super
end

def succ
if (@step)
@prime += @step
@step = 6 - @step
else
case @prime
when 1; @prime = 2
when 2; @prime = 3
when 3; @prime = 5; @step = 2
end
end
@prime
end
alias next succ
def rewind
initialize
end
end

# Internal use. An implementation of prime table by trial division method.
class TrialDivision
include Singleton

def initialize # :nodoc:
# These are included as class variables to cache them for later uses.  If memory
#   usage is a problem, they can be put in Prime#initialize as instance variables.

# There must be no primes between @primes[-1] and @next_to_check.
@primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101]
# @next_to_check % 6 must be 1.
@next_to_check = 103            # @primes[-1] - @primes[-1] % 6 + 7
@ulticheck_index = 3            # @primes.index(@primes.reverse.find {|n|
#   n < Math.sqrt(@@next_to_check) })
@ulticheck_next_squared = 121   # @primes[@ulticheck_index + 1] ** 2
end

# Returns the cached prime numbers.
def cache
@primes
end
alias primes cache
alias primes_so_far cache

# Returns the +index+th prime number.
#
# +index+ is a 0-based index.
def [](index)
while index >= @primes.length
# Only check for prime factors up to the square root of the potential primes,
#   but without the performance hit of an actual square root calculation.
if @next_to_check + 4 > @ulticheck_next_squared
@ulticheck_index += 1
@ulticheck_next_squared = @primes.at(@ulticheck_index + 1) ** 2
end
# Only check numbers congruent to one and five, modulo six. All others

#   are divisible by two or three.  This also allows us to skip checking against
#   two and three.
@primes.push @next_to_check if @primes[2..@ulticheck_index].find {|prime| @next_to_check % prime == 0 }.nil?
@next_to_check += 4
@primes.push @next_to_check if @primes[2..@ulticheck_index].find {|prime| @next_to_check % prime == 0 }.nil?
@next_to_check += 2
end
@primes[index]
end
end

# Internal use. An implementation of Eratosthenes' sieve
class EratosthenesSieve
include Singleton

def initialize
@primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101]
# @max_checked must be an even number
@max_checked = @primes.last + 1
end

def get_nth_prime(n)
compute_primes while @primes.size <= n
@primes[n]
end

private
def compute_primes
# max_segment_size must be an even number
max_segment_size = 1e6.to_i
max_cached_prime = @primes.last
# do not double count primes if #compute_primes is interrupted
# by Timeout.timeout
@max_checked = max_cached_prime + 1 if max_cached_prime > @max_checked

segment_min = @max_checked
segment_max = [segment_min + max_segment_size, max_cached_prime * 2].min
root = Integer.sqrt(segment_max)

segment = ((segment_min + 1) .. segment_max).step(2).to_a

(1..Float::INFINITY).each do |sieving|
prime = @primes[sieving]
break if prime > root
composite_index = (-(segment_min + 1 + prime) / 2) % prime
while composite_index < segment.size do
segment[composite_index] = nil
composite_index += prime
end
end

@primes.concat(segment.compact!)

@max_checked = segment_max
end
end```