La conjetura de Levy

PorJosé A. Alonso 12-marzo-201825-abril-2021

    7 = 3 + 2 x 2
    9 = 3 + 2 x 3 =  5 + 2 x 2
   11 = 5 + 2 x 3 =  7 + 2 x 2
   13 = 3 + 2 x 5 =  7 + 2 x 3
   15 = 3 + 2 x 5 = 11 + 2 x 2
   17 = 3 + 2 x 7 =  7 + 2 x 5 = 11 + 2 x 3 = 13 + 2 x 2
   19 = 5 + 2 x 7 = 13 + 2 x 3

7 = 3 + 2 x 2

9 = 3 + 2 x 3 = 5 + 2 x 2

11 = 5 + 2 x 3 = 7 + 2 x 2

13 = 3 + 2 x 5 = 7 + 2 x 3

15 = 3 + 2 x 5 = 11 + 2 x 2

17 = 3 + 2 x 7 = 7 + 2 x 5 = 11 + 2 x 3 = 13 + 2 x 2

19 = 5 + 2 x 7 = 13 + 2 x 3

y conjeturó que todos los número impares mayores o iguales que 7 se pueden escribir como la suma de un primo y el doble de un primo. El objetivo de los siguientes ejercicios es comprobar la conjetura de Levy.

Definir las siguientes funciones

   descomposicionesLevy :: Integer -> [(Integer,Integer)]
   graficaLevy          :: Integer -> IO ()

1 2	descomposicionesLevy :: Integer -> [(Integer,Integer)] graficaLevy :: Integer -> IO ()

tales que

(descomposicionesLevy x) es la lista de pares de primos (p,q) tales que x = p + 2q. Por ejemplo,

     descomposicionesLevy  7  ==  [(3,2)]
     descomposicionesLevy  9  ==  [(3,3),(5,2)]
     descomposicionesLevy 17  ==  [(3,7),(7,5),(11,3),(13,2)]

descomposicionesLevy 7 == [(3,2)]

descomposicionesLevy 9 == [(3,3),(5,2)]

descomposicionesLevy 17 == [(3,7),(7,5),(11,3),(13,2)]

(graficaLevy n) dibuja los puntos (x,y) tales que x pertenece a [7,9..7+2x(n-1)] e y es el número de descomposiciones de Levy de x. Por ejemplo, (graficaLevy 200) dibuja

Comprobar con QuickCheck la conjetura de Levy.

Soluciones

[schedule on=’2018-03-19′ at=»06:00″]

import Data.Numbers.Primes
import Test.QuickCheck
import Graphics.Gnuplot.Simple

descomposicionesLevy :: Integer -> [(Integer,Integer)]
descomposicionesLevy x =
  [(p,q) | p <- takeWhile (< x) (tail primes)
         , let q = (x - p) `div` 2
         , isPrime q]

-- graficaLevy 300
graficaLevy :: Integer -> IO ()
graficaLevy n =
  plotList [ Key Nothing
           , XRange (7,fromIntegral (7+2*(n-1)))
           , PNG ("La_conjetura_de_Levy-" ++ show n ++ ".png")
           ]
           [(x, length (descomposicionesLevy x)) | x <- [7,9..7+2*(n-1)]] 

-- La propiedad es
prop_Levy :: Integer -> Bool
prop_Levy x =
  not (null (descomposicionesLevy (7 + 2 * abs x)))

-- La comprobación es
--    λ> quickCheck prop_Levy
--    +++ OK, passed 100 tests.

import Data.Numbers.Primes

import Test.QuickCheck

import Graphics.Gnuplot.Simple

descomposicionesLevy :: Integer -> [(Integer,Integer)]

descomposicionesLevy x =

[(p,q) | p <- takeWhile (< x) (tail primes)

, let q = (x - p) `div` 2

, isPrime q]

-- graficaLevy 300

graficaLevy :: Integer -> IO ()

graficaLevy n =

plotList [ Key Nothing

, XRange (7,fromIntegral (7+2*(n-1)))

, PNG ("La_conjetura_de_Levy-" ++ show n ++ ".png")

]

[(x, length (descomposicionesLevy x)) | x <- [7,9..7+2*(n-1)]]

-- La propiedad es

prop_Levy :: Integer -> Bool

prop_Levy x =

not (null (descomposicionesLevy (7 + 2 * abs x)))

-- La comprobación es

-- λ> quickCheck prop_Levy

-- +++ OK, passed 100 tests.

6 Comentarios

import Test.QuickCheck
import Data.Numbers.Primes     (primes, isPrime)
import Graphics.Gnuplot.Simple (plotList, Attribute (Key))

descomposicionesLevy :: Integer -> [(Integer,Integer)]
descomposicionesLevy x =
  [(y,z`div`2)
  | y <- takeWhile (< x) primes,
    let z = x-y,
        even z && isPrime (z`div`2)]

-- Gráfica
-- =======

graficaLevy :: Integer -> IO ()
graficaLevy n =
  plotList
  [Key Nothing]
  [(x,length (descomposicionesLevy x)) | x <- [7,9..7+2*(n-1)]]

-- Propiedad
-- =========

-- La propiedad es
propLevy :: Integer -> Property
propLevy x = x >= 7 && odd x ==> not (null (descomposicionesLevy x))

-- La comprobación es
--    λ> quickCheck propLevy
--    +++ OK, passed 100 tests.

import Test.QuickCheck

import Data.Numbers.Primes (primes, isPrime)

import Graphics.Gnuplot.Simple (plotList, Attribute (Key))

descomposicionesLevy :: Integer -> [(Integer,Integer)]

descomposicionesLevy x =

[(y,z`div`2)

| y <- takeWhile (< x) primes,

let z = x-y,

even z && isPrime (z`div`2)]

-- Gráfica

-- =======

graficaLevy :: Integer -> IO ()

graficaLevy n =

plotList

[Key Nothing]

[(x,length (descomposicionesLevy x)) | x <- [7,9..7+2*(n-1)]]

-- Propiedad

-- =========

-- La propiedad es

propLevy :: Integer -> Property

propLevy x = x >= 7 && odd x ==> not (null (descomposicionesLevy x))

-- La comprobación es

-- λ> quickCheck propLevy

-- +++ OK, passed 100 tests.

Responder

Una solución por recursión

descomposicionesLevy :: Integer -> [(Integer,Integer)]
descomposicionesLevy n = aux n primes
  where aux n (x:xs)
          | n > x && even (n-x) && isPrime a = [(x, a)] ++ aux n xs
          | x < n && odd (n-x)               = aux n xs
          | x < n && not (isPrime a)         = aux n xs
          | otherwise                        = []
          where a = div (n-x) 2
 
graficaLevy :: Integer -> IO ()
graficaLevy n =
  plotList [ Title "Conjetura de Levy"
           , Key Nothing]
           [(x,length(descomposicionesLevy x)) | x <- [7,9..7+2*(n-1)]]
 
prop_Levy :: Integer -> Property
prop_Levy x =
  odd x && x >= 7 ==> not (null (descomposicionesLevy x))

descomposicionesLevy :: Integer -> [(Integer,Integer)]

descomposicionesLevy n = aux n primes

where aux n (x:xs)

| n > x && even (n-x) && isPrime a = [(x, a)] ++ aux n xs

| x < n && odd (n-x) = aux n xs

| x < n && not (isPrime a) = aux n xs

| otherwise = []

where a = div (n-x) 2

graficaLevy :: Integer -> IO ()

graficaLevy n =

plotList [ Title "Conjetura de Levy"

, Key Nothing]

[(x,length(descomposicionesLevy x)) | x <- [7,9..7+2*(n-1)]]

prop_Levy :: Integer -> Property

prop_Levy x =

odd x && x >= 7 ==> not (null (descomposicionesLevy x))

Responder

import Data.Numbers.Primes
import Graphics.Gnuplot.Simple
import Test.QuickCheck

descomposicionesLevy :: Integer -> [(Integer,Integer)]
descomposicionesLevy n =
  [(x,y) | x <- takeWhile (<n) primes
         , let y = (n-x) `div` 2
         , (n-x) `mod` 2 ==0
         , isPrime y]
 
graficaLevy :: Integer -> IO ()
graficaLevy n =
  plotList [Key Nothing]
           [(x,length (descomposicionesLevy x)) | x <- [7,9..7+2*(n-1)]]
 
propLevy :: Integer -> Property
propLevy x = x >= 7 && odd x ==> not (null (descomposicionesLevy x))

import Data.Numbers.Primes

import Graphics.Gnuplot.Simple

import Test.QuickCheck

descomposicionesLevy :: Integer -> [(Integer,Integer)]

descomposicionesLevy n =

[(x,y) | x <- takeWhile (<n) primes

, let y = (n-x) `div` 2

, (n-x) `mod` 2 ==0

, isPrime y]

graficaLevy :: Integer -> IO ()

graficaLevy n =

plotList [Key Nothing]

[(x,length (descomposicionesLevy x)) | x <- [7,9..7+2*(n-1)]]

propLevy :: Integer -> Property

propLevy x = x >= 7 && odd x ==> not (null (descomposicionesLevy x))

Responder

import Data.Numbers.Primes (primes)

descomposicionesLevy :: Integer -> [(Integer,Integer)]
descomposicionesLevy n = [(a,b) | a <- takeWhile (< n) primes
                                , b <- takeWhile  (<= c) primes
                                , n == a + 2*b]
  where c =  div n 2

import Data.Numbers.Primes (primes)

descomposicionesLevy :: Integer -> [(Integer,Integer)]

descomposicionesLevy n = [(a,b) | a <- takeWhile (< n) primes

, b <- takeWhile (<= c) primes

, n == a + 2*b]

where c = div n 2

Responder

-- Conjetura de Levy

-- Librerías usadas

import Data.Numbers.Primes
import Graphics.Gnuplot.Simple
import Data.List
import Test.QuickCheck

descomposicionesLevy :: Integer -> [(Integer,Integer)]
descomposicionesLevy n =
  [(x,y) | x <- [0..n]
         , y <- [0..n]
         , isPrime x
         , isPrime y
         , x+2*y == n]

-- Eficiencia

-- λ> descomposicionesLevy 17
-- [(3,7),(7,5),(11,3),(13,2)]
-- (0.01 secs, 1,332,512 bytes)

-- λ> descomposicionesLevy 487
-- [(5,241),(29,229),(41,223),(89,199),(101,193),(173,157),(233,127),(269,109),(281,103),(293,97),(353,67),(401,43),(449,19),(461,13)]
-- (0.91 secs, 1,489,047,344 bytes)

-- Gráfica

graficaLevy :: Integer -> IO ()
graficaLevy n =
  plotList [Key Nothing]
           [(x,length (descomposicionesLevy x)) | x <- [7,9..7+2*(n-1)]]

-- Propiedad

propLevy :: Integer -> Property
propLevy x = x >= 7 && odd x ==> not (null (descomposicionesLevy x))

-- λ> quickCheck propLevy
-- +++ OK, passed 100 tests.
-- (0.12 secs, 69,287,392 bytes)

-- Conjetura de Levy

-- Librerías usadas

import Data.Numbers.Primes

import Graphics.Gnuplot.Simple

import Data.List

import Test.QuickCheck

descomposicionesLevy :: Integer -> [(Integer,Integer)]

descomposicionesLevy n =

[(x,y) | x <- [0..n]

, y <- [0..n]

, isPrime x

, isPrime y

, x+2*y == n]

-- Eficiencia

-- λ> descomposicionesLevy 17

-- [(3,7),(7,5),(11,3),(13,2)]

-- (0.01 secs, 1,332,512 bytes)

-- λ> descomposicionesLevy 487

-- [(5,241),(29,229),(41,223),(89,199),(101,193),(173,157),(233,127),(269,109),(281,103),(293,97),(353,67),(401,43),(449,19),(461,13)]

-- (0.91 secs, 1,489,047,344 bytes)

-- Gráfica

graficaLevy :: Integer -> IO ()

graficaLevy n =

plotList [Key Nothing]

[(x,length (descomposicionesLevy x)) | x <- [7,9..7+2*(n-1)]]

-- Propiedad

propLevy :: Integer -> Property

propLevy x = x >= 7 && odd x ==> not (null (descomposicionesLevy x))

-- λ> quickCheck propLevy

-- +++ OK, passed 100 tests.

-- (0.12 secs, 69,287,392 bytes)

Responder

import Graphics.Gnuplot.Simple
import Test.QuickCheck

descomposicionesLevy :: Integer -> [(Integer, Integer)]
descomposicionesLevy x =
  [(p,q)| p <- ys
        , q <- ys
        , p + 2*q == x]
  where ys = take (fromIntegral x) primos
 
primos :: [Integer]
primos = [p | p <- [2..], numDiv p == 2] 
 
numDiv :: Integer -> Int
numDiv n = length[x | x <- [1..n], n `mod` x == 0]
 
graficaLevy :: Integer -> IO ()
graficaLevy n =
  plotList [Key Nothing]
           [(x,length (descomposicionesLevy x)) | x <- [7,9..7+2*(n-1)]]
 
propiedadLevy :: Integer -> Property
propiedadLevy n =
  n >= 7 && not (even n) ==> not (length (descomposicionesLevy n) == 0)

import Graphics.Gnuplot.Simple

import Test.QuickCheck

descomposicionesLevy :: Integer -> [(Integer, Integer)]

descomposicionesLevy x =

[(p,q)| p <- ys

, q <- ys

, p + 2*q == x]

where ys = take (fromIntegral x) primos

primos :: [Integer]

primos = [p | p <- [2..], numDiv p == 2]

numDiv :: Integer -> Int

numDiv n = length[x | x <- [1..n], n `mod` x == 0]

graficaLevy :: Integer -> IO ()

graficaLevy n =

plotList [Key Nothing]

[(x,length (descomposicionesLevy x)) | x <- [7,9..7+2*(n-1)]]

propiedadLevy :: Integer -> Property

propiedadLevy n =

n >= 7 && not (even n) ==> not (length (descomposicionesLevy n) == 0)

Responder

La conjetura de Levy

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