# Euler problems/131 to 140

(Difference between revisions)

## 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
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:

```import List

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..]]

dign x=(+1)\$floor\$logBase 10\$fromInteger x

extEuclid x1 x2 x3 y1 y2 1=(x3,y2)
extEuclid x1 x2 x3 y1 y2 y3
=extEuclid y1 y2 y3 t1 t2 t3
where
(k,t3)=divMod x3 y3
t1=x1-k*y1
t2=x2-k*y2
-- find a ,b
-- mod (x*a) y = 1
-- mod (y*b) x = 1
congrue x y |x>y= extEuclid 1 0 x 0 1 y
|otherwise =(a,b)
where
(b,a)=congrue y x
--fastfun 7=1219
fastfun x
|x==1=0
|p1>1000000=0
|otherwise= a*d+p1
where
p1=primes!!x
p2=primes!!(x+1)
dp=p2-p1
d=10^dign p1
dmod=mod d p2
eu=(+p2)\$fst\$congrue dmod p2
a=mod (eu*dp) p2

groups=1000

funsum k=sum[fastfun a|
a<-[1+k*groups..groups*(k+1)]
]

-- write file to change bignum to small num
=if (num>79)
then return()
else do appendFile "file.log" \$(show\$funsum num)  ++"  "++(show num) ++"\n"
-- first use main to make file.log
-- then run problem_134

split :: Char -> String -> [String]
split = unfoldr . split'

split' :: Char -> String -> Maybe (String, String)
split' c l
| null l = Nothing
| otherwise = Just (h, drop 1 t)
where (h, t) = span (/=c) l

problem_134=do
let y=sum\$map sToInt \$lines x
print  y```

## 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
]
-- write file to change bignum to small num
=if (num>49)
then return()
else do appendFile "file.log" ((show\$length\$problem_136 num)  ++ "\n")

## 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 =
m<-[1..10],
n<-[1..10],
n*n==d*m*m+1
]
findmin_s d =
m<-[1..10],
n<-[1..10],
n*n==d*m*m-1
]
findmu d y=
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 =
m<-[1..10],
n<-[1..10],
n*n==d*m*m+1
]
findmin_s d =
m<-[1..10],
n<-[1..10],
n*n==d*m*m+1
]
findmu d y=
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)]```