# Euler problems/131 to 140

## Problem 131

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

Solution:

```
problem_131 = undefined
```

## Problem 132

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

Solution:

```
problem_132 = undefined
```

## Problem 133

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

Solution:

```
problem_133 = undefined
```

## Problem 134

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

Solution:

```
problem_134 = undefined
```

## 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]
```

## 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
```

## Problem 137

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

Solution:

```
problem_137 = undefined
```

## Problem 138

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

Solution:

```
problem_138 = undefined
```

## Problem 139

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

Solution:

```
problem_139 = undefined
```

## Problem 140

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

Solution:

```
problem_140 = undefined
```