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Euler problems/131 to 140

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1 Problem 131

Determining primes, p, for which n3 + n2p is a perfect cube.

Solution:

primes=sieve [2..]
sieve (x:xs)=x:sieve [y|y<-xs,mod y x>0]
primeFactors n = factor n primes
    where
        factor _ [] = []
        factor m (p:ps) | p*p > m        = [m]
                        | m `mod` p == 0 = p : factor (m `div` p) (p:ps)
                        | otherwise      = factor m ps
 
isPrime n = case (primeFactors n) of
                (_:_:_)   -> False
                _         -> True
problem_131 =
    length $ takeWhile (<1000000) 
    [x|
    a<-[1 .. ],
    let x=(3*a*(a+1)+1),
    isPrime x]

2 Problem 132

Determining the first forty prime factors of a very large repunit.

Solution:

-- primes powMod in problem_133
fun x = 
    (powMod x 10 n)==1
    where
    n=10^9
--add 3
p132 =sum$take 41 [a|a<-primes,fun a]
problem_132 =p132-3

3 Problem 133

Investigating which primes will never divide a repunit containing 10n digits.

Solution:

import Data.List
mulMod :: Integral a => a -> a -> a -> a
mulMod a b c= (b * c) `rem` a
squareMod :: Integral a => a -> a -> a
squareMod a b = (b * b) `rem` a
pow' :: (Num a, Integral b) => (a -> a -> a) -> (a -> a) -> a -> b -> a
pow' _ _ _ 0 = 1
pow' mul sq x' n' = f x' n' 1
    where
    f x n y
        | n == 1 = x `mul` y
        | r == 0 = f x2 q y
        | otherwise = f x2 q (x `mul` y)
        where
            (q,r) = quotRem n 2
            x2 = sq x
powMod :: Integral a => a -> a -> a -> a
powMod m = pow' (mulMod m) (squareMod m)
 
merge xs@(x:xt) ys@(y:yt) = case compare x y of
    LT -> x : (merge xt ys)
    EQ -> x : (merge xt yt)
    GT -> y : (merge xs yt)
 
diff  xs@(x:xt) ys@(y:yt) = case compare x y of
    LT -> x : (diff xt ys)
    EQ -> diff xt yt
    GT -> diff xs yt
 
primes, nonprimes :: [Integer]
primes    = [2,3,5] ++ (diff [7,9..] nonprimes) 
nonprimes = foldr1 f . map g $ tail primes
    where f (x:xt) ys = x : (merge xt ys)
          g p = [ n*p | n <- [p,p+2..]]
fact25 m  
    | m `mod` 2 == 0 = 2 : fact25 (m `div` 2) 
    | m `mod` 5 == 0 = 5 : fact25 (m `div` 5)
    | otherwise      = []
fun x
    |n==x-1=True
    |otherwise= (powMod x 10 n)==1
    where
    n=product$fact25 (x-1)
--miss 2 3 5
test =sum$takeWhile (<100)[a|a<-primes,not$fun a]
p133 =sum$takeWhile (<100000)[a|a<-primes,not$fun a]
problem_133 = p133+2+3+5

4 Problem 134

Finding the smallest positive integer related to any pair of consecutive primes.

Solution:

merge xs@(x:xt) ys@(y:yt) = case compare x y of
    LT -> x : (merge xt ys)
    EQ -> x : (merge xt yt)
    GT -> y : (merge xs yt)
 
diff  xs@(x:xt) ys@(y:yt) = case compare x y of
    LT -> x : (diff xt ys)
    EQ -> diff xt yt
    GT -> diff xs yt
 
primes, nonprimes :: [Integer]
primes    = [2,3,5] ++ (diff [7,9..] nonprimes) 
nonprimes = foldr1 f . map g $ tail primes
    where f (x:xt) ys = x : (merge xt ys)
          g p = [ n*p | n <- [p,p+2..]]
 
problem_134 :: Integer
problem_134 = sum xs
           where
             ps = drop 2 primes
             ds = takeWhile ((< 10^6) . fst) $ zip ps (tail ps)
             xs = map (uncurry func) ds
 
 
expo :: Integer -> Int
expo = length . show
 
find :: Integer -> Integer -> (Integer, Integer)
find a b = findStep a 1 0 b 0 1
 
findStep :: Integer -> Integer -> Integer -> Integer -> Integer
                             -> Integer -> (Integer, Integer)
findStep a x1 x2 b y1 y2 =
   case divMod a b of
        (q,0) -> (x2, y2)
        (q,r) -> findStep b x2 (x1-q*x2) r  y2 (y1-q*y2)
 
checkL :: Integer -> Integer -> (Integer,Integer)
checkL 0 _ = (-1,1)
checkL n d
    = let (u,v) = find n d
      in if u <= 0 then (n-v,d+u) else (-v,u)
 
func :: Integer -> Integer -> Integer
func p1 p2
    = n*p2
      where
        md = 10^(expo p1)
        (_,h) = checkL p2 md
        n = p1*h `mod` md

5 Problem 135

Determining the number of solutions of the equation x2 − y2 − z2 = n.

Solution:

import Control.Monad
import Data.List
import Data.Array.ST
import Control.Monad.ST
import Control.Monad.Cont
 
--  ghc -package mtl p135.hs
p135 m = runST (do
  counts <- newArray (1,m-1) 0 :: ST s (STUArray s Int Int)
 
  forM_ [1 .. m - 1] $ \ x ->
    forM_' [x `div` 3 + 1 .. div m 2] $ \ break n ->
 
        let t = (n + x) * (3 * n - x)
        in if t < m
             then lift $ incArray counts t
             else break ()
 
  xs <- getElems counts
  return $ length $ filter (==10) xs)
 
  where
    forM_' xs f = flip runContT return $ callCC $ forM_ xs . f
 
    incArray arr index = do
      v <- readArray arr index
      writeArray arr index (v + 1)
main=appendFile "p135.log"$show $p135 (10^6)
problem_135=main

6 Problem 136

Discover when the equation x2 − y2 − z2 = n has a unique solution.

Solution:

import List
find2km :: Integral a => a -> (a,a)
find2km n = f 0 n
    where 
        f k m
            | r == 1 = (k,m)
            | otherwise = f (k+1) q
            where (q,r) = quotRem m 2        
 
millerRabinPrimality :: Integer -> Integer -> Bool
millerRabinPrimality n a
    | a <= 1 || a >= n-1 = 
        error $ "millerRabinPrimality: a out of range (" 
              ++ show a ++ " for "++ show n ++ ")" 
    | n < 2 = False
    | even n = False
    | b0 == 1 || b0 == n' = True
    | otherwise = iter (tail b)
    where
        n' = n-1
        (k,m) = find2km n'
        b0 = powMod n a m
        b = take (fromIntegral k) $ iterate (squareMod n) b0
        iter [] = False
        iter (x:xs)
            | x == 1 = False
            | x == n' = True
            | otherwise = iter xs
 
pow' :: (Num a, Integral b) => (a -> a -> a) -> (a -> a) -> a -> b -> a
pow' _ _ _ 0 = 1
pow' mul sq x' n' = f x' n' 1
    where 
        f x n y
            | n == 1 = x `mul` y
            | r == 0 = f x2 q y
            | otherwise = f x2 q (x `mul` y)
            where
                (q,r) = quotRem n 2
                x2 = sq x
 
mulMod :: Integral a => a -> a -> a -> a
mulMod a b c = (b * c) `mod` a
squareMod :: Integral a => a -> a -> a
squareMod a b = (b * b) `rem` a
powMod :: Integral a => a -> a -> a -> a
powMod m = pow' (mulMod m) (squareMod m)
isPrime x=foldl   (&& )True [millerRabinPrimality x y|y<-[2,3,7,61]]
nextPrime x=head [a|a<-[(x+1)..],isPrime a]
lazyPrimeSieve :: [Integer] -> [Integer]
lazyPrimeSieve [] = []
lazyPrimeSieve (x:xs) = x : (lazyPrimeSieve $ filter (\y -> y `rem` x /= 0) xs)
 
oddPrimes :: [Integer]
oddPrimes = lazyPrimeSieve [3,5..]
fun =2+sum[testPrime a|a<-takeWhile (<100) oddPrimes]
limit=50000000
groups=1000000
fillmap num total rlimit=do
    let a=nextPrime num
    if a>rlimit then 
        return  total 
        else do
            let b=testPrime a 
            fillmap (a+1) (total+b) rlimit
testPrime p =p1+p2+p3
    where
    p1=if mod p 4==3 then 1 else 0
    p2=if p*4<limit then 1 else 0
    p3=if p*16<limit then 1 else 0
problem_136 b=fillmap ming 0 maxg
    where
    ming=b*groups
    maxg=(b+1)*groups
google num
-- write file to change bignum to small num
  =if (num>49)
      then return()
      else do t1<-problem_136 num
              appendFile "file.log" $show t1  ++ "\n"
              google (num+1)
main=do
    appendFile "file.log" $show fun  ++ "\n"
    k<-fillmap 100 0 groups
    appendFile "file.log" $show k  ++ "\n"
    google 1
sToInt =(+0).read 
problem_136a=do
    s<-readFile "file.log"
    print$sum$map sToInt$lines s

7 Problem 137

Determining the value of infinite polynomial series for which the coefficients are Fibonacci numbers.

Solution:

-- afx=x/(1-x-x^2)=n
--   ->5*n^2+2*n+1=d^2
--   ->let k=10*n+2
--   ->20*d^2=k^2+16
--   ->5*d^2=k^2+4
--   ->let d k is even
--   ->5*d^2=k^2+1
--   ->let d k is odd
--   ->5*d^2=k^2+4
 
import Data.List
findmin d = 
    d:head [[n,m]|
    m<-[1..10],
    n<-[1..10],
    n*n==d*m*m+1
    ]
findmin_s d =
    d:head [[n,m]|
    m<-[1..10],
    n<-[1..10],
    n*n==d*m*m-1 
    ]
findmu d y= 
    d:head [[n,m]|
    m<-[1..10],
    n<-[1..10],
    n*n==d-y*m
    ]
mux2 [d,a, b]=[d,a,-b]
mult [d,a, b] [_,a1, b1]=
    d:[a*a1+d*b*b1,a*b1+b*a1]
pow 1 x=x
pow n x =mult x $pow (n-1) x 
    where
    mult [d,a, b] [_,a1, b1]=d:[a*a1+d*b*b1,a*b1+b*a1]
fun =[c|
    a<-[1..20],
    [_,b,_]<-powmu a,
    let bb=abs(b),
    mod bb 5==1,
    let c=div bb 5
    ]
powmu n =
    [a,b]
    where
    c=pow n $findmin 5
    x1=findmu 5 4
    x2=mux2 x1
    a=mult c x1 
    b=mult c x2
fun2=[c|
    a<-[1..20], 
    let[_,b,_]=pow a $findmin_s 5,
    let bb=b*2,mod bb 5==1,
    let c=div bb 5
    ]
problem_137 =(!!14)$sort $(++fun)$fun2

8 Problem 138

Investigating isosceles triangle for which the height and base length differ by one.

Solution:

{-
 - 4*m^2-16*m*n-4*n^2+1=0
 - 4*m^2-16*m*n-4*n^2-1=0
 - (m-2*n)^2-5*n^2=1
 - (m-2*n)^2-5*n^2=-1
 -}
import Data.List
mult [d,a, b] [_,a1, b1]=
    d:[a*a1+d*b*b1,a*b1+b*a1]
pow 1 x=x
pow n x =mult x $pow (n-1) x 
    where
    mult [d,a, b] [_,a1, b1]=d:[a*a1+d*b*b1,a*b1+b*a1]
-- 2^2-5*1^2=-1
-- so [5,2,1]
fun =
    [d^2+c^2|
    a<-[1..20],
    let [_,b,c]=pow a [5,2,1],
    let d=2*c+b
    ]
-- 9^2-5*4^2=1
-- so [5,9,4]
fun2 =
    [d^2+c^2|
    a<-[1..20],
    let [_,b,c]=pow a [5,9,4],
    let d=2*c+b
    ]
problem_138 =sum$take 12 $nub$sort (fun++fun2)

9 Problem 139

Finding Pythagorean triangles which allow the square on the hypotenuse square to be tiled.

Solution:

{-
 -                              2                        2
 -                     (n - 1) y  - 2 n x y + (- n - 1) x  = 0
 --->
 -                                           2       2       2
 -                          ((n - 1) y - n x)  = (2 n  - 1) x
 --->
 -                       2        2
 -                    2 n  - 1 = k
 -
 -}
import Data.List
mult [d,a, b] [_,a1, b1]=
    d:[a*a1+d*b*b1,a*b1+b*a1]
pow 1 x=x
pow n x =mult x $pow (n-1) x 
div2 [x,y,z]
    |mod x 2==0 && mod y 2==0 && mod z 2==0=
        [div x 2,div y 2,div z 2]
    |otherwise=[x,y,z]
-- 1^2-2*1^2=-1
-- so [2,1,1]
fun =map div2 [
    side
    |a<-[3,5..40],
    let [_,k,n]=pow a [2,1,1],
    let m=lcm (n+k) (n-1),
    let x=div m (n+k),
    let y=div m (n-1),
    let side=[y^2-x^2, 2*x*y, y^2+x^2]
    ]
limit=100000000
problem_139=sum [div limit$sum a|a<-fun,  sum a<limit]

10 Problem 140

Investigating the value of infinite polynomial series for which the coefficients are a linear second order recurrence relation.

Solution:

{-
                                      2
                                   3 x  + x
                            agx= ------------
                                    2
                                 - x  - x + 1
--->
                                   2             2
                                5 n  + 14 n + 1=d
 
--->                          k = 10 n + 14
--->                          20*d^2=k^2-176
--->                          k = 5 n + 2
--->                          5*d^2=k^2-44
-}
import Data.List
findmin d = 
    d:head [[n,m]|
    m<-[1..10],
    n<-[1..10],
    n*n==d*m*m+1
    ]
findmin_s d = 
    d:head [[n,m]|
    m<-[1..10],
    n<-[1..10],
    n*n==d*m*m+1
    ]
findmu d y= 
    d:head [[n,m]| 
    m<-[1..10],
    n<-[1..10],
    n*n==d+y*m
    ]
mux2 [d,a, b]=[d,a,-b]
mult [d,a, b] [_,a1, b1]=d:[a*a1+d*b*b1,a*b1+b*a1]
div2 [d,a, b] =d:[div a 2,div b 2]
pow 1 x=x
pow n x =mult x $pow (n-1) x 
fun =
    [c|
    a<-[1..20],
    [_,b,_]<-powmu a,
    let bb=abs(b),
    mod bb 5==2,
    let c=div bb 5
    ]
fun2=
    [c|
    a<-[1..20],
    [_,b,_]<-powmu1 a ,
    let bb=(abs b)*2,
    mod bb 5==2,
    let c=div bb 5
    ]
powmu n =
    [a,b,a1,a2,b1,b2]
    where
    c=pow n $findmin 5
    x1=findmu 5 44
    x2=mux2 x1
    a=mult c x1 
    b=mult c x2
    a1=div2$mult a [5,3, -1]
    a2=div2$mult a [5,3, 1]
    b1=div2$mult b [5,3, -1]
    b2=div2$mult b [5,3, 1]
powmu1 n =
    [a,b]
    where
    c=pow n $findmin_s 5
    x1=findmu 5 11
    x2=mux2 x1
    a=mult c x1 
    b=mult c x2
problem_140 =sum $take 30 [a-1|a<-nub$sort (fun++fun2)]