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Euler problems/131 to 140

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Solution:
 
Solution:
 
<haskell>
 
<haskell>
problem_137 = undefined
+
-- 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
 
</haskell>
 
</haskell>
   

Revision as of 01:54, 18 December 2007

Contents

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:

problem_132 = undefined

3 Problem 133

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

Solution:

problem_133 = undefined

4 Problem 134

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

Solution:

problem_134 = undefined

5 Problem 135

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

Solution:

import List
primes :: [Integer]
primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
 
primeFactors :: Integer -> [Integer]
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 :: Integer -> Bool
isPrime 1 = False
isPrime n = case (primeFactors n) of
                (_:_:_)   -> False
                _         -> True
fstfac x = [(head a ,length a)|a<-group$primeFactors x]
fac [(x,y)]=[x^a|a<-[0..y]]
fac (x:xs)=[a*b|a<-fac [x],b<-fac xs]
factors x=fac$fstfac x
fastfun x
    |mod x 4==3=[a|a<-factors x,a*a<3*x]
    |mod x 16==4=[a|let n=div x 4,a<-factors n,a*a<3*n]
    |mod x 16==12=[a|let n=div x 4,a<-factors n,a*a<3*n]
    |mod x 16==0=[a|let n=div x 16,a<-factors n,a*a<3*n]
    |otherwise=[]
 
slowfun x =[a|a<-factors x,a*a<3*x,let b=div x a,mod (a+b) 4==0]
 
problem_135 =[a|a<-[1..groups],(length$fastfun a)==10]

6 Problem 136

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

Solution:

-- fastfun in the problem 135
groups=1000000
pfast=[a|a<-[1..5000],(length$fastfun a)==1]
pslow=[a|a<-[1..5000],(length$slowfun a)==1]
-- find len pfast=len pslow+2
-- so sum file.log and +2
problem_136 b=[a|a<-[1+b*groups..groups*(b+1)],(length$fastfun a)==1]
google num
-- write file to change bignum to small num
  =if (num>49)
      then return()
      else do appendFile "file.log" ((show$length$problem_136 num)  ++ "\n")
              google (num+1)
main=google 0

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:

problem_138 = undefined

9 Problem 139

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

Solution:

problem_139 = undefined

10 Problem 140

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

Solution:

problem_140 = undefined