# Euler problems/131 to 140

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## Revision as of 09:04, 14 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:

problem_137 = undefined

## 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