46 lines
1.7 KiB
Text
46 lines
1.7 KiB
Text
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Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of positive numbers less than or equal to n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6.
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The number 1 is considered to be relatively prime to every positive number, so φ(1)=1.
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Interestingly, φ(87109)=79180, and it can be seen that 87109 is a permutation of 79180.
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Find the value of n, 1 < n < 107, for which φ(n) is a permutation of n and the ratio n/φ(n) produces a minimum.
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---
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From problem 069 we have a fast way to compute the function phi.
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> import Data.List
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> import Prime
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> phi :: Int -> Int
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> phi n = let decomp = map head $ group $ primeFactors n
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> in foldl' (\acc p-> (acc `div` p) * (p-1)) n decomp
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We also define nphi which is just n/phi(n)
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> nphi :: Int -> Float
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> nphi n = (fromIntegral n) / fromIntegral (phi n)
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We also need a function to verify a number is a permutation of another number.
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> arePermutEquiv p q = sort (show p) == sort (show q)
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Now we only have to verify for all n
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> interrestingNumbers n = concatMap foo [2..n]
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> where
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> foo x = if arePermutEquiv x (phi x)
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> then [(x,nphi x)]
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> else []
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The function to return the result:
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> takeMaximal :: [(Int,Float)] -> (Int,Float) -> (Int,Float)
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> takeMaximal [] (best,minratio) = (best,minratio)
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> takeMaximal ((n,ratio):xs) (best,minratio) =
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> if ratio<minratio then takeMaximal xs (n,ratio) else takeMaximal xs (best,minratio)
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> main = do
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> print $ takeMaximal ( interrestingNumbers (10^7) ) (0,10^7)
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