## Feature #14383

### Making prime_division in prime.rb Ruby 3 ready.

**Description**

I have been running old code in Ruby 2.5.0 (released 2017.12.25) to check for

speed and compatibility. I still see the codebase in `prime.rb`

hardly has

changed at all (except for replacing `Math.sqrt`

with `Integer.sqrt`

).

To achieve the Ruby 3 goal to make it at least three times faster than Ruby 2

there are three general areas where Ruby improvements can occur.

- increase the speed of its implementation at the machine level
- rewrite its existing codebase in a more efficient|faster manner
- use faster algorithms to implement routines and functions

I want to suggest how to address the later two ways to improve performance of

specifically the `prime_division`

method in the `prime.rb`

library.

I've raised and made suggestions to some of these issues here

ruby-issues forum and now hope to invigorate additional discussion.

Hopefully with the release of 2.5.0, and Ruby 3 conceptually closer to reality,

more consideration will be given to coding and algorithmic improvements to

increase its performance too.

**Mathematical correctness**

First I'd like to raise what I consider *math bugs* in `prime_division`

, in how

it handles `0`

and `-1`

inputs.

> -1.prime_division => [[-1,1]] > 0.prime_division Traceback (most recent call last): 4: from /home/jzakiya/.rvm/rubies/ruby-2.5.0/bin/irb:11:in `<main>' 3: from (irb):85 2: from /home/jzakiya/.rvm/rubies/ruby-2.5.0/lib/ruby/2.5.0/prime.rb:30:in `prime_division' 1: from /home/jzakiya/.rvm/rubies/ruby-2.5.0/lib/ruby/2.5.0/prime.rb:203:in `prime_division' ZeroDivisionError (ZeroDivisionError)

First, `0`

is a perfectly respectable integer, and is non-prime, so its output should be `[]`

,

an empty array to denote it has no prime factors. The existing behavior is solely a matter of

`prime_division`

's' implementation, and does not take this mathematical reality into account.

The output for `-1`

is also mathematically wrong because `1`

is also non-prime (and correctly

returns `[]`

), well then mathematically so should `-1`

. Thus, `prime_division`

treats `-1`

as

a new prime number, and factorization, that has no mathematical basis. Thus, for mathematical

correctness and consistency `-1`

and `0`

should both return `[]`

, as none have prime factors.

> -1.prime_division => [] > 0.prime_division => [] > 1.prime_division => []

There's a very simple one-line fix to `prime_division`

to do this:

# prime.rb class Prime def prime_division(value, generator = Prime::Generator23.new) -- raise ZeroDivisionError if value == 0 ++ return [] if (value.abs | 1) == 1

**Simple Code and Algorithmic Improvements**

As stated above, besides the machine implementation improvements, the other

areas of performance improvements will come from coding rewrites and better

algorithms. Below is the coding of `prime_division`

. This coding has existed at

least since Ruby 2.0 (the farthest I've gone back).

# prime.rb class Integer # Returns the factorization of +self+. # # See Prime#prime_division for more details. def prime_division(generator = Prime::Generator23.new) Prime.prime_division(self, generator) end end class Prime 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 end

This can be rewritten in more modern and idiomatic Ruby, to become much shorter

and easier to understand.

require 'prime.rb' class Integer def prime_division1(generator = Prime::Generator23.new) Prime.prime_division1(self, generator) end end class Prime def prime_division1(value, generator = Prime::Generator23.new) # raise ZeroDivisionError if value == 0 return [] if (value.abs | 1) == 1 pv = value < 0 ? [[-1, 1]] : [] value = value.abs generator.each do |prime| count = 0 while (value1, mod = value.divmod(prime); mod) == 0 value = value1 count += 1 end pv.push [prime, count] unless count == 0 break if prime > value1 end pv.push [value, 1] if value > 1 pv end end

By merely rewriting it we get smaller|concise code, that's easier to understand,

which is slightly faster. A *triple win!* Just paste the above code into a 2.5.0

terminal session, and run the benchmarks below.

def tm; s=Time.now; yield; Time.now-s end n = 500_000_000_000_000_000_008_244_213; tm{ pp n.prime_division } [[3623, 1], [61283, 1], [352117631, 1], [6395490847, 1]] => 27.02951016 n = 500_000_000_000_000_000_008_244_213; tm{ pp n.prime_division1 } [[3623, 1], [61283, 1], [352117631, 1], [6395490847, 1]] => 25.959149721

Again, we get a *triple win* to this old codebase by merely rewriting it. It can

be made 3x faster by leveraging the `prime?`

method from the `OpenSSL`

library to

perform a more efficient|faster factoring algorithm, and implementation.

require 'prime.rb' require 'openssl' class Integer def prime_division2(generator = Prime::Generator23.new) return [] if (self.abs | 1) == 1 pv = self < 0 ? [-1] : [] value = self.abs prime = generator.next until value.to_bn.prime? or value == 1 while prime (pv << prime; value /= prime; break) if value % prime == 0 prime = generator.next end end pv << value if value > 1 pv.group_by {|prm| prm }.map{|prm, exp| [prm, exp.size] } end end

Here we're making much better use of Ruby idioms and libraries (`enumerable`

and

`openssl`

), leading to a much greater performance increase. A bigger *triple win*.

Pasting this code into a 2.5.0 terminal session gives the following results.

# Hardware: System76 laptop; I7 cpu @ 3.5GHz, 64-bit Linux def tm; s=Time.now; yield; Time.now-s end n = 500_000_000_000_000_000_008_244_213; tm{ pp n.prime_division } [[3623, 1], [61283, 1], [352117631, 1], [6395490847, 1]] => 27.02951016 n = 500_000_000_000_000_000_008_244_213; tm{ pp n.prime_division1 } [[3623, 1], [61283, 1], [352117631, 1], [6395490847, 1]] => 25.959149721 n = 500_000_000_000_000_000_008_244_213; tm{ pp n.prime_division2 } [[3623, 1], [61283, 1], [352117631, 1], [6395490847, 1]] => 9.39650374

`prime_division2`

is much more usable for significantly larger numbers and use

cases than `prime_division`

. I can even do multiple times better than this, if

you review the above cited forum thread.

My emphasis here is to show there are a lot of possible *low hanging fruit*

performance gains ripe for the picking to achieve Ruby 3 performance goals, if we

look (at minimum) for simpler|better code rewrites, and then algorithmic upgrades.

So the question is, are the devs willing to upgrade the codebase to provide the

demonstrated performance increases shown here for `prime_division`

?

**Files**

### History

#### Updated by shevegen (Robert A. Heiler) over 1 year ago

I won't go into all points since your issue makes even my issue

requests seem small. :-)

It may be easier to split your suggestions into separate issues

though.

For example, the 0 and -1 situation, if it is a bug (probably

is but I have not checked the official math definition for

prime divisions myself yet), it may be better to split it into

another issue and detach it from your other statements made,

e. g. the code quality or the speedup gain from optimizing

the ruby code.

I don't think that the ruby core devs are against smaller

improvements at all, irrespective of the 3.x goal (which I

think will be achieved via the JIT/mjit anyway) but it

may be simpler to have smaller issues. Some of the bigger

issues take longer to resolve. At any rate, that is just my

opinion - feel free to ignore it. :)

You could perhaps also add the overall discussion to:

https://bugs.ruby-lang.org/projects/ruby/wiki/DevelopersMeeting20180124Japan

If an attendee notices the issue here (and if it is worth

discussing; I think you indirectly also pointed out that

some parts of the ruby core/stdlib do not receive equal

attention possibly due to a lack of maintainers; I think

that one problem may also be that many ruby hackers would

not even know which area of core/stdlib may need improvements

or attention. Not just the prime division situation you

described, but also e. g. improvements to the cgi-part of

ruby, even if these are minor - it's not easy to know which

areas of standard distributed ruby need improvements.)

#### Updated by mrkn (Kenta Murata) over 1 year ago

Currently, `prime_division`

can factorize any negative integers that are less than -1 like:

[2] pry(main)> -12.prime_division => [[-1, 1], [2, 2], [3, 1]]

Do you think how to treat these cases?

I think raising Math::DomainError is better for 1, 0, and any negative integers cases.

#### Updated by jzakiya (Jabari Zakiya) over 1 year ago

The major problem with `prime_division`

trying to accommodate negative numbers is

that, mathematically, prime factorization is really only considered over positive integers > 1.

I understand the creators intent, to be able to reconstruct negative integers from their prime factorization,

but that's not what's done mathematically. `1|0`

are not primes or composites so they can't be factored

(they have no factors). You can see the Numberphile video explanation of this, or if you prefer, the wikipedia one.

From a serious mathematical perspective, `prime_division`

(and the whole

`prime.rb`

lib) is inadequate for doing real high-level math. This is why I created

the primes-utils gem, so I could do fast math involving `primes`

. I also wrote the

Primes-Utils Handbook, to provide and explain all the gem's source code.

The Coureutils library, a part of all `[Li|U]nix`

systems, provides a world class factorization function

factor. You're not going to create a Ruby version that will come close to it. I use `factor`

for

my default factoring function. Here's an implementation that mimics `prime_division`

.

class Integer def factors(p=0) # p is unused variable for method consistency return [] if self | 1 == 1 factors = self < 0 ? [-1] : [] factors += `factor #{self.abs}`.split(' ')[1..-1].map(&:to_i) factors.group_by {|prm| prm }.map {|prm, exp| [prm, exp.size] } end end

And here's the performance difference against `prime_division2`

, the fastest

version of the previous functions.

n = 500_000_000_000_000_000_008_244_213; tm{ pp n.prime_division2 } [[3623, 1], [61283, 1], [352117631, 1], [6395490847, 1]] => 9.39650374 n = 500_000_000_000_000_000_008_244_213; tm{ pp n.factors } [[3623, 1], [61283, 1], [352117631, 1], [6395490847, 1]] => 0.007200317

If it were up to me, I would deprecate the whole `prime.rb`

library and use the

capabilities in `primes-utils`

, to give Ruby a much more useful|fast prime math

library. In fact, I'm doing a serious rewrite of it for version 3.0, using faster

math, with faster implementations. When Ruby ever gets true parallel capabilities

it'll be capable of making use of it.

But again in general, there are articles|videos showing how to speed up Ruby code.

There are projects like Fast Ruby , specifically devoted to identifying and categorizing

specific code constructs for speed. These resources can be used for evaluating the core

codebase to help rewrite it using the fastest coding constructs.

Ruby is 20+ years old now, and I would imagine some (a lot of?) code has probably

never been reviewed|considered for rewriting for performance gains. A lot of

similarly aged languages have gone through this process (Python, Perl, PHP) and

now it's Ruby's turn.

So while it's natural to talk and focus on the future machine implementation of

Ruby 3, I think you can be getting current language speedups by making the

ongoing codebase as efficiently and performantly written now, which will

translate to an even faster Ruby 3.

This is also a way to get more than just the C guru devs involved in creating

Ruby 3. I wouldn't mind evaluating existing libraries for simplicity|speed

rewrites if I knew my work would be truly considered. This would provide casual

users an opportunity to learn more of the internal workings of Ruby, while

contributing to its development, which can only be to Ruby's benefit. How about

a `Library of the Week|Month`

or `GoSC`

rewrite project, or a `Make Ruby Faster Now`

project! Lots of ways to make this effort fun and interesting.

#### Updated by graywolf (Gray Wolf) over 1 year ago

Unless I copy-pasted wrong your `prime_division2`

is /significantly/ slower for small numbers:

$ ruby bm.rb prime_division - current in prime.rb prime_division_2 - proposed version in #14383 by jzakiya `factor` - spawning coreutils' factor command `factor` is left out from the first benchmark, spawning 1_000_000 shells proves nothing. 1_000_000 times of 10 user system total real prime_division 1.496456 0.000059 1.496515 ( 1.496532) prime_division_2 104.094586 6.469827 110.564413 (110.565201) 1_000 times of 2**256 user system total real prime_division 0.069389 0.000000 0.069389 ( 0.069391) prime_division_2 0.325075 0.000000 0.325075 ( 0.325088) `factor` 0.163589 0.073234 0.992784 ( 1.021447) 10 times of 500_000_000_000_000_000_008_244_213 user system total real prime_division 328.625069 0.033017 328.658086 (328.801649) prime_division_2 118.690491 0.000119 118.690610 (118.691372) `factor` 0.002487 0.000030 0.024145 ( 0.025040)

script:

```
require 'prime'
require 'openssl'
require 'benchmark'
class Integer
def prime_division_2(generator = Prime::Generator23.new)
return [] if (self.abs | 1) == 1
pv = self < 0 ? [-1] : []
value = self.abs
prime = generator.next
until value.to_bn.prime? or value == 1
while prime
(pv << prime; value /= prime; break) if value % prime == 0
prime = generator.next
end
end
pv << value if value > 1
pv.group_by {|prm| prm }.map{|prm, exp| [prm, exp.size] }
end
end
def factor(num)
return [] if num | 1 == 1
factors = num < 0 ? [-1] : []
factors += `factor #{num.abs}`.split(' ')[1..-1].map(&:to_i)
factors.group_by {|prm| prm }.map {|prm, exp| [prm, exp.size] }
end
puts <<~EOF
prime_division - current in prime.rb
prime_division_2 - proposed version in #14383 by jzakiya
`factor` - spawning coreutils' factor command
`factor` is left out from the first benchmark, spawning 1_000_000
shells proves nothing.
EOF
puts
puts "1_000_000 times of 10"
puts
Benchmark.bm(16) do |x|
x.report('prime_division') { 1_000_000.times { 10.prime_division } }
x.report('prime_division_2') { 1_000_000.times { 10.prime_division_2 } }
end
puts
puts "1_000 times of 2**256"
puts
Benchmark.bm(16) do |x|
num = 2**256
x.report('prime_division') { 1_000.times { num.prime_division } }
x.report('prime_division_2') { 1_000.times { num.prime_division_2 } }
x.report('`factor`') { 1_000.times { factor(num) } }
end
puts
puts "10 times of 500_000_000_000_000_000_008_244_213"
puts
Benchmark.bm(16) do |x|
num = 500_000_000_000_000_000_008_244_213
x.report('prime_division') { 10.times { num.prime_division } }
x.report('prime_division_2') { 10.times { num.prime_division_2 } }
x.report('`factor`') { 10.times { factor(num) } }
end
```

#### Updated by jzakiya (Jabari Zakiya) over 1 year ago

Well, I did say "serious" math, didn't I.

2.5.0 :097 > 2**256 => 115792089237316195423570985008687907853269984665640564039457584007913129639936 2.5.0 :099 > n = 2**256 + 1; tm{ pp n.factors } [[1238926361552897, 1], [93461639715357977769163558199606896584051237541638188580280321, 1]] => 11.187528889 2.5.0 :103 > n = 2**256 + 6; tm{ pp n.factors } [[2, 1], [9663703905367, 1], [5991082217089035545953414273093775102416031327093273407023490613, 1]] => 181.896643599 2.5.0 :105 > n = 2**256 + 7; tm{ pp n.factors } [[92243, 1], [14633710594132193, 1], [39071613028785859, 1], [2195480924803008082289717129761953851423, 1]] => 86.285821283 2.5.0 :107 > n = 2**256 + 8; tm{ pp n.factors } [[2, 3], [3, 1], [683, 1], [4049, 1], [85009, 1], [2796203, 1], [31797547, 1], [81776791273, 1], [2822551529460330847604262086149015242689, 1]] => 210.062944465 2.5.0 :109 > n = 2**256 + 9; tm{ pp n.factors } [[5, 1], [37181, 1], [210150995838577, 1], [2963851002430239530676411809410149856603062505748058817897, 1]] => 8.70362184

#### Updated by mame (Yusuke Endoh) over 1 year ago

Your `prime_division2`

uses `OpenSSL::BN#prime?`

. You may know, it is a Miller-Rabin *probabilistic* primality test which may cause a false positive. In short, I suspect that your code may (very rarely) return a wrong result. Am I right? If so, it is unacceptable.

In my personal opinion, `lib/prime.rb`

is not for practical use, but just for fun. The speed is not so important for `lib/prime.rb`

. So I think it would be good to keep it idyllic.

BTW, I've released a gem namely faster_prime, a faster substitute for `lib/prime.rb`

.

prime_division_current - current in prime.rb prime_division_jzakiya - proposed version in #14383 by jzakiya prime_division_mame - https://github.com/mame/faster_prime 1_000_000 times of 10 user system total real prime_division_current 1.203975 0.000000 1.203975 ( 1.204458) prime_division_jzakiya 50.366586 8.994353 59.360939 ( 59.369606) prime_division_mame 1.031790 0.000000 1.031790 ( 1.031963) 1_000 times of 2**256 user system total real prime_division_current 0.057273 0.000219 0.057492 ( 0.057495) prime_division_jzakiya 0.230237 0.000000 0.230237 ( 0.230288) prime_division_mame 0.074282 0.000106 0.074388 ( 0.074459) 10 times of 500_000_000_000_000_000_008_244_213 user system total real prime_division_current 222.418914 0.032561 222.451475 (222.491855) prime_division_jzakiya 76.686295 0.000191 76.686486 ( 76.694430) prime_division_mame 0.102406 0.000000 0.102406 ( 0.102408)

$ time ruby -rfaster_prime -e 'p (2**256+1).prime_division' [[1238926361552897, 1], [93461639715357977769163558199606896584051237541638188580280321, 1]] real 2m13.676s user 2m13.645s sys 0m0.008s

:-)

My gem is written in pure Ruby (not uses OpenSSL), but not so simple, so I have no intention to replace the standard `lib/prime.rb`

, though.

#### Updated by jzakiya (Jabari Zakiya) over 1 year ago

Hi Yusuke.

Ah, we agree, `prime.rb`

is not conducive for doing heavy-duty math. :-)

Please look at and play with my primes-utils gem.

It has a minimal universal useful set of methods for doing prime math.

Again, I'm in the process of rewriting it and adding more methods. Maybe you

want to collaborate with me and give Ruby a much better|serious prime math library.

Ruby is a serious language and deserves much better in areas of scientific and

numerical computation. This need has been recognized by such projects as

SciRuby, and I would like Ruby 3 to be better in these areas.

I installed your `faster_prime`

gem and ran it on my laptop, and started

looking at the code. I haven't thoroughly studied it yet, but may I make

some suggestions to simplify and speed it up.

I don't know if you knew, but starting with Ruby 2.5, it now has `Integer.sqrt`

.

We had a vigorous discussion on this issue here.

So you can replace your code below:

module Utils module_function FLOAT_BIGNUM = Float::RADIX ** (Float::MANT_DIG - 1) # Find the largest integer m such that m <= sqrt(n) def integer_square_root(n) if n < FLOAT_BIGNUM Math.sqrt(n).floor else # newton method a, b = n, 1 a, b = a / 2, b * 2 until a <= b a = b + 1 a, b = b, (b + n / b) / 2 until a <= b a end end

with `Integer.sqrt`

. BTW, here's a fast|accurate pure Ruby implementation of it.

class Integer def sqrt # Newton's method version used in Ruby for Integer#sqrt return nil if (n = self) < 0 # or however you want to handle this case return n if n < 2 b = n.bit_length x = 1 << (b-1)/2 | n >> (b/2 + 1) # optimum initial root estimate while (t = n / x) < x; x = ((x + t) >> 1) end x end end

Also in same file, you can do this simplification:

def mod_sqrt(a, prime) return 0 if a == 0 case when prime == 2 a.odd? ? 1 : 0 => a & 1 => or better => a[0] # lsb of number ....

Finally, in `prime_factorization.rb`

we can simplify and speedup code too. Don't

count the primes factors when you find then, just stick then in an array, then

when finished with factorization process you can use `Enumerable#group_by`

to

format output of them in one step. Saves code size|complexity, and is faster.

I also restructured the code like mine to make it simpler. Now you don't have

to do so many tests, and you get rid of all those individual `yields`

, which

complicates and slow things down. You will have to modify the part of your

code that uses `prime_factorization`

, but that code should be simpler|faster too.

require "faster_prime/utils" module FasterPrime module PrimeFactorization module_function # Factorize an integer def prime_factorization(n) return enum_for(:prime_factorization, n) unless block_given? return if n == 0 if n < 0 yield [-1, 1] n = -n end SMALL_PRIMES.each do |prime| if n % prime == 0 c = 0 begin n /= prime c += 1 end while n % prime == 0 yield [prime, c] end if prime * prime > n yield [n, 1] if n > 1 return end return if n == 1 end if PrimalityTest.prime?(n) yield [n, 1] else d = nil until d [PollardRho, MPQS].each do |algo| begin d = algo.try_find_factor(n) rescue Failed else break end end end pe = Hash.new(0) prime_factorization(n / d) {|p, e| pe[p] += e } prime_factorization(d) {|p, e| pe[p] += e } pe.keys.sort.each do |p| yield [p, pe[p]] end end end end

Change to below, with appropriate changes for factoring algorithm inside loop.

require "faster_prime/utils" module FasterPrime module PrimeFactorization module_function # Factorize an integer def prime_factorization(n) return enum_for(:prime_factorization, n) unless block_given? # 'factors' will hold the number of individual prime factors return [] if n.abs | 1 == 1 # for n = -1, 0, or 1 factors = n < 0 ? [-1] | [] # if you feel compelled for negative nums n = n.abs SMALL_PRIMES.each do |prime| # extract small prime factors, if any (factors << prime; n /= prime) while n % prime == 0 end # at this point n is either a prime, 1, or a composite of non-small primes until PrimalityTest.prime?(n) or n == 1 # exit when n is 1 or prime # if you're in here then 'n' is a composite thats needs factoring # when you find a factor, stick it in 'factors' and reduce 'n' by it # ultimately 'n' will be reduced to a prime or 1 # do whatever needs to be done to make this work right d = nil until d [PollardRho, MPQS].each do |algo| begin d = algo.try_find_factor(n) rescue Failed else break end end end end # at this point 'n' is either a prime or 1 factors << n if n > 1 # stick 'n' in 'factors' if it's a prime # 'factors' now has all the number of individual prime factors # now use Enumerable#group_by to make life simple and easy :-) # the xx.sort is unnecessary if you find the prime factors sequentially factors.group_by(&:itself).sort.map { |prime, exponents| [prime, exponents.size] } end end

This is now a generic template for factorization. To upgrade just use better|faster

`PrimalityTest`

and `factorization`

functions. You also see it has separated each

distinct algorithmic function into single area of concern, in a conceptually more

functional programming style. This is now also able to be possibly implemented in

parallel because of the isolation of functional areas of concern.

#### Updated by jzakiya (Jabari Zakiya) over 1 year ago

Also, FYI, to go along with using `Integer.sqrt`

, you can save some code (and increase performance)

using the OpenSSL library, which already has some of the methods in `utils.rb`

.

require 'openssl' mod_pow(a,b,n) => a.to_bn.mod_exp(b,n) mod_inv(a,n) => a.to_bn.mod_inverse(n)

#### Updated by yugui (Yuki Sonoda) over 1 year ago

**Target version**set to*Next Major***Assignee**set to*yugui (Yuki Sonoda)*

Faster is better -- with certain conditions. Considering those conditions, I would like to take the following approach.

- Remove the dependency from
`mathn.rb`

to`prime.rb`

- Move
`prime.rb`

from`lib/`

to a bundled gem - Apply some of the patches.

This is because of the following conditions I would like the change to satisfy.

- Simplicity of the dependency graph in the standard library
- I don't want to let
`mathn`

have too much dependency.

- I don't want to let
- Keeping the focus of the standard library
- We are moving more libraries from
`lib/`

to bundled gems. The solution must be consistent to this trend.

- We are moving more libraries from
- Certain degree of backward compatibility
- the algorithm must be deterministic.

#### Updated by jzakiya (Jabari Zakiya) over 1 year ago

Thank you for taking this project into consideration.

FYI, there is C open source code for both the `Coreutils`

factor function and the APR-CL deterministic

primality test. However, Miller-Rabin is faster, and can be implemented as deterministic, at least up

to 25-digits, so maybe a good `Integer#prime?`

method should be a hybrid combination, depending on number size.

The documentation for the APR-CL code says it's good up to 6021-digits on 64-bit systems

(not that anyone should expect to process those size numbers in any reasonable times).

To make sense of all these possibilities, a determination of the max number size|range has

to be made, since we are talking about an infinite set of numbers (primes). I wouldn't

think Ruby (or any general language) would have a primes library that will find the

largest Mersenne Primes, for example. However, with the right math and implementation,

thousands of digits numbers can be easily worked with within reasonable times and memory usage.

As an example, below is output of my `Integer#primesmr`

method, for numerating primes

within a number range, here for 100, 1000, and 2000 digit numbers. It combines the `M-R`

test with my math techniques using `prime generators`

to identify a minimum set of

`prime candidates`

within the ranges, and then checking their primality.

This is part of my primes-utils gem, which has a Primes-Utils Handbook,

which explains all the math, and provides the full gem source code.

It so far has the methods `primes|mr|f`

, `primescnt|mr|f`

, `nthprime`

, `prime|mr?`

and

`factors|prime_division`

in current version 2.7. For the (next) 3.0 release I'm adding

`next_prime`

and `prev_prime`

too. I think these are a good (minimum) set of general

purpose methods that provide the majority of information most people would seek in

a primes library. I'll probably create single hybrid methods also where now I have multiple

versions. I'm playing with JRuby for using parallel algorithms I already have implemented

in C++/Nim until CRuby can provide capability.

Again, thanks for the Ruby team agreeing to take on this effort.

2.5.0 :017 > n= (10**100+267); tm{ p n.primesmr n+5000 } [10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000267, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000949, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001243, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001293, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001983, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002773, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002809, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002911, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002967, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003469, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003501, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003799, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004317, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004447, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004491] => 0.06310091 2.5.0 :018 > n= (10**1000+267); tm{ p n.primesmr n+5000 } [10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000453, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001357, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002713, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004351] => 11.089284953 2.5.0 :019 > n= (10**2000+267); tm{ p n.primesmr n+5000 } [100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004561] => 56.129938705 2.5.0 :020 >

#### Updated by jzakiya (Jabari Zakiya) over 1 year ago

FYI, I re-ran the examples above (and an additional one) using the `Miller-Rabin`

implementation in my `primes-utls`

3.0 development branch. Not only is it

deterministic up to about 25-digits, but it's way faster too. It can probably be

made faster by using a `WITNESS_RANGES`

list with fewer witnesses, and can now

be easily extended as optimum witnesses are found for larger number ranges.

I provide this to raise the issue that absolute determinism for a primality test

function maybe shouldn't be an absolute criteria for selection. Of course, it is

the ideal, but sometimes `striving for perfection can be the enemy of good enough`

.

Again, hybrid methods may be able to combine the best of all worlds.

> n= (10**100+267); tm{ p n.primesmr n+5000 } => 0.056422744 3.0 dev => 0.06310091 2.7 > n= (10**1000+267); tm{ p n.primesmr n+5000 } => 7.493292636 3.0 dev => 11.089284953 2.7 > n= (10**2000+267); tm{ p n.primesmr n+5000 } => 36.614877874 3.0 dev => 56.129938705 2.7 > n= (10**3000+267); tm{ p n.primesmr n+5000 } => 10000000000........1027 => 192.866720666 3.0 dev => 286.244139793 2.7

# Miller-Rabin version in Primes-Utils 2.7 def primemr?(k=20) # increase k for more reliability n = self.abs return true if [2,3].include? n return false unless [1,5].include?(n%6) and n > 1 d = n - 1 s = 0 (d >>= 1; s += 1) while d.even? k.times do a = 2 + rand(n-4) x = a.to_bn.mod_exp(d,n) # x = (a**d) mod n next if x == 1 or x == n-1 (s-1).times do x = x.mod_exp(2,n) # x = (x**2) mod n return false if x == 1 break if x == n-1 end return false if x != n-1 end true # n is prime (with high probability) end

# Miller-Rabin version in Primes-Utils 3.0 dev # Returns true if +self+ is a prime number, else returns false. def primemr? primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43] return primes.include? self if self <= primes.last return false unless primes.reduce(:*).gcd(self) == 1 wits = WITNESS_RANGES.find {|range, wits| range > self} # [range, [wit_prms]] or nil witnesses = wits && wits[1] || primes witnesses.each {|p| return false unless miller_rabin_test(p) } true end private # Returns true if +self+ passes Miller-Rabin Test on witness +b+ def miller_rabin_test(b) # b is a witness to test with n = d = self - 1 d >>= 1 while d.even? y = b.to_bn.mod_exp(d, self) # x = (b**d) mod n until d == n || y == n || y == 1 y = y.mod_exp(2, self) # y = (y**2) mod self d <<= 1 end y == n || d.odd? end WITNESS_RANGES = { 2_047 => [2], 1_373_653 => [2, 3], 25_326_001 => [2, 3, 5], 3_215_031_751 => [2, 3, 5, 7], 2_152_302_898_747 => [2, 3, 5, 7, 11], 3_474_749_660_383 => [2, 3, 5, 7, 11, 13], 341_550_071_728_321 => [2, 3, 5, 7, 11, 13, 17], 3_825_123_056_546_413_051 => [2, 3, 5, 7, 11, 13, 17, 19, 23], 318_665_857_834_031_151_167_461 => [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37], 3_317_044_064_679_887_385_961_981 => [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41] }

#### Updated by jzakiya (Jabari Zakiya) over 1 year ago

This version is probably better, as it's a little faster, and takes a complete

list of witnesses for a given number, which can be determined separately.

# Returns true if +self+ is a prime number, else returns false. def primemr? primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43] return primes.include? self if self <= primes.last return false unless primes.reduce(:*).gcd(self) == 1 wits = WITNESS_RANGES.find {|range, wits| range > self} # [range, [wit_prms]] or nil witnesses = wits && wits[1] || primes miller_rabin_test(witnesses) end private # Returns true if +self+ passes Miller-Rabin Test on witness +b+ def miller_rabin_test(witnesses) # use witness list to test with neg_one_mod = n = d = self - 1 d >>= 1 while d.even? witnesses.each do |b| s = d y = b.to_bn.mod_exp(d, self) # y = (b**d) mod self until s == n || y == 1 || y == neg_one_mod y = y.mod_exp(2, self) # y = (y**2) mod self s <<= 1 end return false unless y == neg_one_mod || s.odd? end true end