How to resolve the algorithm Arithmetic/Rational step by step in the Haskell programming language

Published on 7 June 2024 03:52 AM

How to resolve the algorithm Arithmetic/Rational step by step in the Haskell programming language

Table of Contents

Problem Statement

Create a reasonably complete implementation of rational arithmetic in the particular language using the idioms of the language.

Define a new type called frac with binary operator "//" of two integers that returns a structure made up of the numerator and the denominator (as per a rational number). Further define the appropriate rational unary operators abs and '-', with the binary operators for addition '+', subtraction '-', multiplication '×', division '/', integer division '÷', modulo division, the comparison operators (e.g. '<', '≤', '>', & '≥') and equality operators (e.g. '=' & '≠'). Define standard coercion operators for casting int to frac etc. If space allows, define standard increment and decrement operators (e.g. '+:=' & '-:=' etc.). Finally test the operators: Use the new type frac to find all perfect numbers less than 219 by summing the reciprocal of the factors.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Arithmetic/Rational step by step in the Haskell programming language

This Haskell code snippet calculates and prints the first N perfect numbers. Perfect numbers are positive integers that are equal to the sum of their proper divisors, excluding the number itself.

The main function:

  • Defines a constant n to specify how many perfect numbers to print (in this case, 4).
  • Uses the mapM_ function to print each perfect number found using take.
  • take takes the first n elements from the list generated by the list comprehension [ candidate | ... ].
  • The list comprehension generates candidate perfect numbers by iterating through a range of integers [2 .. 2 ^ 19] and filtering for numbers that satisfy the condition getSum candidate == 1, which checks if the sum of their divisors (excluding the number itself) is 1.

The getSum function:

  • Calculates the sum of the proper divisors of a candidate perfect number.
  • It adds 1 % candidate, which represents the number itself, and then sums over all factors (2 up to the square root of the candidate) that evenly divide the candidate.
  • For each factor, it adds 1 % factor and 1 % (candidate div factor), which represent the factor and its corresponding pair in the factorization of the candidate.

Source code in the haskell programming language

import Data.Ratio ((%))

-- Prints the first N perfect numbers.
main = do
  let n = 4
  mapM_ print $
    take
      n
      [ candidate
      | candidate <- [2 .. 2 ^ 19] 
      , getSum candidate == 1 ]
  where
    getSum candidate =
      1 % candidate +
      sum
        [ 1 % factor + 1 % (candidate `div` factor)
        | factor <- [2 .. floor (sqrt (fromIntegral candidate))] 
        , candidate `mod` factor == 0 ]


  

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