Ruby on Rails

"primes-utils" is a Rubygem which provides a suite of extremely fast  (relative to Ruby's standard library) utility methods for testing and  generating primes.    Install it the usual way:        $ gem install primes-utils    Then require inside ruby as:        > require 'primes/utils'    The suite consists of the following methods:    1)prime?  Determine if the absolute value of an integer is prime.  Return 'true' or 'false'.  This replaces the `prime?` method  in the `prime.rb` standard library.    2)primemr?(k=20)  Determine if the absolute value of an integer is prime using Miller-Rabin test.  Return 'true' or 'false'.  Miller-Rabin here is super fast, but probabilistic (not deterministic), primality test.  https://en.wikipedia.org/wiki/Miller-Rabin_primality_test  The method's reliability can be increased by increasing the default input parameter of k=20.    3)factors(p=13) or prime_division(p=13)  Determine the prime factorization of the absolute value of an integer.  This replaces the `prime division` method in the `prime.rb` standard library.  Returns an array of arrays of factors and exponents: [[2,4],[3,2],[5,1]] => (2^4)(3^2)(5^1) = (16)(9)(5) = 720  Default Strictly Prime (SP) Prime Generator (PG) used here is P13.  Can change SP PG used on input. Acceptable primes range (3 - 19).    4)primes(start_num=0)  Create an array of primes from the absolute value range (|start_num| - |end_num|).  The order of the range doesn't matter if both given:  start.primes end <=> end.prime start  If only one parameter used, then all the primes upto that number will be returned.    5)primenth(p=11) or nthprime(p=11)  Return the value of the nth (absolute value) prime.  Default Strictly Prime (SP) Prime Generator (PG) used here is P11.  Can change SP PG used on input. Acceptable primes range (3 - 19).    Coding Implementations  The methods `primemr`, `nthprime/primenth`, and `primes` are coded in  pure ruby. The methods `prime?` and `prime_division/factors` have two  implementations. Each has a pure ruby implementation, and also a hybrid  implementation which uses the Unix cli command `factor` if its available  on the host OS. `factor` [5] is an extremely fast C coded factoring  algorithm, part of the GNU Core Utilities package [4].    Upon loading, the gem tests if the desired `factor` command exists on  the host OS. If so, it wraps a system call to it and uses it for  `prime?` and `prime_division/factors`. If not, it uses a fast pure ruby  implementation for each method based on the Sieve of Zakiya (SoZ)[1][2][3].    References  [1]https://www.scribd.com/doc/150217723/Improved-Prim...  [2]https://www.scribd.com/doc/228155369/The-Segmented...  [3]https://www.scribd.com/doc/73385696/The-Sieve-of-Zakiya  [4]https://en.wikipedia.org/wiki/GNU_Core_Utilities  [5]https://en.wikipedia.org/wiki/Factor_(Unix)    Author  Jabari Zakiya