# Solutions for Problem 108

Given the equation (1/x)+(1/y)=n where x,y,n are integers, find the first n with more than 1000 distinct solutions.

Runtime: 956.868 ms

```{- Prime numbers as usual -}
primes :: [Integer]
primes = 2:3:primes'
where
1:p:candidates  = [6*k+r | k <- [0..], r <- [1,5]]
primes'         = p : filter isPrime candidates
isPrime n       = all (not . divides n) \$ takeWhile (\p -> p*p <= n) primes'
divides n p     = n `mod` p == 0

{- Primorials as usual -}
primorials = map (\x -> product (take x primes)) [1..]

{- Numbers to check.
- Note that the primorials (and miltiples of) have the most solutions. -}
pots = inner 1 2 (tail primorials)
where
inner m cp pps@(p:ps)
| mp < p = mp : inner (m + 1) cp pps
| otherwise = inner 1 p ps
where
mp = m * cp

{- The number of solutions for a given "n" -}
solns n = length (filter (\x -> mod (n * x) (x - n) == 0) [n+1..2*n])

{- Find the first "n" with over 1000 solutions. -}
main = print (head (filter (\x -> solns x > 1000) pots))
```