How to resolve the algorithm Pythagorean triples step by step in the Ruby programming language

Published on 12 May 2024 09:40 PM
#Ruby

How to resolve the algorithm Pythagorean triples step by step in the Ruby programming language

Table of Contents

Problem Statement

A Pythagorean triple is defined as three positive integers

( a , b , c )

{\displaystyle (a,b,c)}

where

a < b < c

{\displaystyle a<b<c}

, and

a

2

b

2

=

c

2

.

{\displaystyle a^{2}+b^{2}=c^{2}.}

They are called primitive triples if

a , b , c

{\displaystyle a,b,c}

are co-prime, that is, if their pairwise greatest common divisors

g c d

( a , b )

g c d

( a , c )

g c d

( b , c )

1

{\displaystyle {\rm {gcd}}(a,b)={\rm {gcd}}(a,c)={\rm {gcd}}(b,c)=1}

. Because of their relationship through the Pythagorean theorem, a, b, and c are co-prime if a and b are co-prime (

g c d

( a , b )

1

{\displaystyle {\rm {gcd}}(a,b)=1}

).   Each triple forms the length of the sides of a right triangle, whose perimeter is

P

a + b + c

{\displaystyle P=a+b+c}

.

The task is to determine how many Pythagorean triples there are with a perimeter no larger than 100 and the number of these that are primitive.

Deal with large values.   Can your program handle a maximum perimeter of 1,000,000?   What about 10,000,000?   100,000,000? Note: the extra credit is not for you to demonstrate how fast your language is compared to others;   you need a proper algorithm to solve them in a timely manner.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Pythagorean triples step by step in the Ruby programming language

Class Definition: The code defines a class named PythagoranTriplesCounter that generates Pythagorean triples up to a specified limit and counts them.

Instance Variables:

  • @limit: The upper bound for the perimeter of the generated Pythagorean triples.
  • @total: The total number of Pythagorean triples within the limit.
  • @primitives: The number of primitive Pythagorean triples (i.e., triples with no common factors).

Constructor:

  • initialize(limit): Initializes the PythagoranTriplesCounter with the desired perimeter limit.

Public Attributes:

  • total: Returns the total number of Pythagorean triples generated within the limit.
  • primitives: Returns the number of primitive Pythagorean triples generated.

Private Methods:

  • generate_triples(a, b, c): Generates Pythagorean triples by iteratively applying the following formulas:
    • (a-2*b+2*c, 2*a-b+2*c, 2*a-2*b+3*c)
    • (a+2*b+2*c, 2*a+b+2*c, 2*a+2*b+3*c)
    • (-a+2*b+2*c, -2*a+b+2*c, -2*a+2*b+3*c)

Main Program:

  • Initializes a perimeter variable perim with a value of 10.
  • Creates a loop that continues until perim exceeds 100,000,000.
  • Inside the loop:
    • Creates a PythagoranTriplesCounter instance with the current perimeter limit.
    • Prints the perimeter, total number of triples, and number of primitive triples generated.
    • Multiplies perim by 10 to increase the perimeter limit for the next iteration.

Source code in the ruby programming language

class PythagoranTriplesCounter
  def initialize(limit)
    @limit = limit
    @total = 0
    @primitives = 0
    generate_triples(3, 4, 5)
  end
  attr_reader :total, :primitives
  
  private
  def generate_triples(a, b, c)
    perim = a + b + c
    return if perim > @limit

    @primitives += 1
    @total += @limit / perim

    generate_triples( a-2*b+2*c, 2*a-b+2*c, 2*a-2*b+3*c)
    generate_triples( a+2*b+2*c, 2*a+b+2*c, 2*a+2*b+3*c)
    generate_triples(-a+2*b+2*c,-2*a+b+2*c,-2*a+2*b+3*c)
  end
end

perim = 10
while perim <= 100_000_000 
  c = PythagoranTriplesCounter.new perim
  p [perim, c.total, c.primitives]
  perim *= 10
end

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