See http://www.zerocater.com/challenge/#mathy-challenge for details of what this does

zerocater_mathy_challenge.rb

# See http://www.zerocater.com/challenge/#mathy-challenge for details of what
# this code does
#
# The approach I took was to Google 'primal sums' and get most of the solution 
# directly:
#
#   http://oeis.org/A007506
#
# I also use pre-computed lists of primes to avoid having to implement a sieve
# in order to get lists of primes.
#
# This code uses the above knowledge of the last digit in the 'fifth chain' sequence
# to get the fifth chain without having to directly compute it. See below.
#
# Requires a system with the 'unzip' command in the path. This code was executed on 
# a standard machine running Ubuntu 12.10 in ~10 seconds (including I/O).

require 'open-uri'

def read_digit_list(url, lines_to_drop)
  puts "Downloading file at URL #{url}"

  primes_list_txt = open(url) { |f| f.read }

  puts "Done."

  # get rid of the header/introductory lines
  primes_list_pruned = primes_list_txt.split("\n").drop(lines_to_drop).join("\n")

  primes_list_pruned.scan(/\d+/).collect(&:to_i)
end

def chain_search(digits, min_last)
  digit_list = []
  sum = 0
  digits.each do |digit|
    digit_list << digit

    sum += digit
    last_digit = digit_list.last
    if sum % last_digit == 0 && last_digit > min_last
      return digit_list.length
    end
  end
end

small_digits = read_digit_list("http://primes.utm.edu/lists/small/100000.txt", 3)
puts chain_search(small_digits, 5) # third chain (already known)
puts chain_search(small_digits, 71) # fourth chain (last digit 369119)

# To find the fifth chain length, we only have to find the position of the digit
# 415074643 (which we got in the Sloane integer sequence above) in a list of 
# primes found on the same server as the 100,000 list above. It's in the 23 
# million range. It's stored as a zip file on the UTM server, so we'll download
# the zip, extract it locally, then read it as a file.

target_digit = 415_074_643
offset = 22_000_000

puts "Downloading 22-23 million primes list"
big_prime_url = "http://primes.utm.edu/lists/small/millions/primes23.zip"
puts "Done. Writing and extracting the file"
zip_download = open(big_prime_url) { |f| f.read }
File.open('primes23.zip', 'w') {|f| f.write(zip_download) }
`unzip primes23.zip`
puts "Done."

big_digit_list = read_digit_list('primes23.txt', 2)

index = big_digit_list.index(target_digit)
final_length = offset + index
puts final_length