Step by Step solution about How to resolve the algorithm Partition an integer x into n primes step by step in the Haskell programming language

This Haskell code implements a function to find all possible partitions of a given number x into n prime numbers. A partition is a way of dividing a number into smaller parts, in this case, prime numbers.

Here's a detailed explanation of the code:

1. Importing Modules:

   • Data.List provides functions for list manipulation, such as delete and intercalate.
   • Data.Numbers.Primes provides access to the list of prime numbers, and primes returns an infinite list of prime numbers.
   • Data.Bool provides the bool function, which takes three arguments and returns either the first or second argument based on the truthiness of the third argument.

2. partitions Function:

   • This is the main function that finds all possible partitions of x into n prime numbers. It takes two integer arguments, x and n.
   • If n is less than or equal to 1, it means there are no partitions, and it returns a list with x as the only element.
   • Otherwise, it calls the go function with the list of prime numbers less than or equal to x, x, and n.

3. go Function:

   • This function recursively finds all possible partitions. It takes three arguments:
     • ps: A list of prime numbers less than or equal to x.
     • x: The number to partition.
     • n: The number of partitions.
   • If n is equal to 1, it means we have found a partition. It checks if x is in the list of prime numbers ps and returns it as a list if it is.
   • Otherwise, it uses the found function to check if x can be partitioned into n prime numbers using the prime numbers in ps.

4. found Function:

   • This function checks if there exists a prime number p in ps such that x - p can be partitioned into n - 1 prime numbers using the remaining prime numbers in ps.
   • It returns True if such a partition exists and False otherwise.

5. main Function:

   • This is the main entry point of the program.
   • It defines a list of test cases as pairs of numbers (x, n).
   • It uses the mapM_ function to apply the putStrLn function to each test case.
   • The putStrLn function prints a string followed by a newline.

6. justifyLeft Function:

   • This is a helper function that justifies a given string s to the left with n characters, using the character c.

import Data.List (delete, intercalate)
import Data.Numbers.Primes (primes)
import Data.Bool (bool)

-------------------- PRIME PARTITIONS ---------------------
partitions :: Int -> Int -> [Int]
partitions x n
  | n <= 1 =
    [ x
    | x == last ps ]
  | otherwise = go ps x n
  where
    ps = takeWhile (<= x) primes
    go ps_ x 1 =
      [ x
      | x `elem` ps_ ]
    go ps_ x n = ((flip bool [] . head) <*> null) (ps_ >>= found)
      where
        found p =
          ((flip bool [] . return . (p :)) <*> null)
            ((go =<< delete p . flip takeWhile ps_ . (>=)) (x - p) (pred n))

-------------------------- TEST ---------------------------
main :: IO ()
main =
  mapM_ putStrLn $
  (\(x, n) ->
      intercalate
        " -> "
        [ justifyLeft 9 ' ' (show (x, n))
        , let xs = partitions x n
          in bool
               (tail $ concatMap (('+' :) . show) xs)
               "(no solution)"
               (null xs)
        ]) <$>
  concat
    [ [(99809, 1), (18, 2), (19, 3), (20, 4), (2017, 24)]
    , (,) 22699 <$> [1 .. 4]
    , [(40355, 3)]
    ]

------------------------- GENERIC -------------------------
justifyLeft :: Int -> Char -> String -> String
justifyLeft n c s = take n (s ++ replicate n c)