{The first consecutive 10-digit prime number in the value of e} .com

!macOS-11.1!ruby-2.7.2p137

Preface (Introduction)

I would like to solve the other issue, Google's classified ads, with Ruby.

Google recruit

What is Google Job Ads? This is the picture above. Translated into Japanese, it is {the first consecutive 10-digit prime number in the value of $ e $} .com.

Apparently, Google in the old days advertised such a unique job advertisement to hire a great IT engineer.

Now, let's solve it using Ruby.

Ruby

Read the value of e

$ e $ is called the Napier number, and its definition is

Expressed as a convergence sequence, $ e = \ lim_ {x \ to \ infinity} \ left (1 + \ frac {1} {x} \ right) ^ x $

Expressed in calculus, $ e = \ sum_ {n = 0} ^ \ infinty \ frac {1} {n!} $.

$ e $ is an irrational number, but this time we will handle up to 200 digits.

2.71828182845904523536028747135266249775
7247093699959574966967627724076630353547
5945713821785251664274274663919320030599
2181741359662904357290033429526059563073
81323286279434907632338298807531952510190

Save this value in exp.txt so that it loads every time you run the code.

def read_exp 
  exp = File.read('exp.txt')
  return exp.delete('.')
end

Reference: Number of Napiers & # x2013; Wikipedia

Primality test

Create a function to determine if the argument n is a prime number.

For n to be a prime number, it must be indivisible by all numbers from 2 to n-1. Therefore, find the remainder of numbers from 2 to n-1 and create a function called a prime number if all of them are not 0.

def prime? n
  for i in 2..n-1 do
    return false if n % i == 0
  end
  return true
end

However, this would be a wasteful calculation, so the upper limit of iteration is $ \ sqrt {n} + 1 $.

Here's an explanation of why this is done.

In the figure below, green represents the first quadrant of $ xy = 6 $ and red represents $ y = x $. At this time, the green line is axisymmetric with the red line as the axis. In other words, with the intersection $ (x, y) = (\ sqrt {6}, \ sqrt {6}) $ of the two graphs as the boundary, the order is reversed, but the same numbers are multiplied.
In that case, we only need to check half, so the range for primality test should be 2 ~ $ \ sqrt {n} + 1 $.

def prime? n
  i = 2
  return true if n == 2
  return false if n % i == 0
  i += 1
  max = Integer.sqrt(n) + 1
  while i <= max do
    return false if n % i == 0
    i += 2
  end
  return true
end

Code

Now, combine the existing functions to complete the program.

exp_prime.rb

def read_exp
  exp = File.read('exp.txt')
  return exp.delete('.')
end

def prime? n
  i = 2
  return true if n == 2
  return false if n % i == 0
  i += 1
  max = Integer.sqrt(n) + 1
  while i <= max do
    return false if n % i == 0
    i += 2
  end
  return true
end

require "./assert_equal"
def test
  expected = [2,3,5,7]
  results = []
  [*2..10].each do |num|
    results << num if prime?(num)
  end
  assert_equal(expected, results)
end

def main
  exp = read_exp()
  for i in 0..exp.length-10 do
    next if exp[i] == '0'
    exp10 = exp[i..i+9].to_i
    if prime?(exp10)
      puts exp10
      break
    end
  end
end

if $PROGRAM_NAME == __FILE__
  main()
end

If you move this

> ruby exp_prime.rb
-> 7427466391

Superbly, it returned a prime number.

Similar

Congratulations. You've made it to level 2. Go to www.Linux.org and enter Bobsyouruncle as the login and the answer to this equation as the password.

f(1)=7182818284
f(2)=8182845904
f(3)=8747135266
f(4)=7427466391
f(5)=__________

Since the value of f (4) is equal to the value given earlier, it seems likely that this is also the value of $ e $. However, I don't know what else they have in common.

Now, let's create a program based on that information.

google_recruit.rb

require './exp_prime'

def main
  exp = read_exp()
  cnt = 0
  for i in 0..exp.length-10 do
    sum = 0
    for j in 0..9 do
      sum += exp[i+j].to_i
    end
    if sum ==49
      cnt += 1
      puts "f(" + cnt.to_s + ") = " + exp[i..i+9]
    end
  end
end

if $PROGRAM_NAME == __FILE__
  main()
end

The output of this program is

f(1) = 7182818284
f(2) = 8182845904
f(3) = 8747135266
f(4) = 7427466391
f(5) = 5966290435
f(6) = 2952605956

Therefore, the answer is 5966290435.

Reference material

• Google recruit problem, exp and prime -Famous Google Job Advertisement / Napier Number Value and 10 Digit Prime Number