Question

It turns out that 12 cm is the smallest length of wire that can be bent to form an integer sided right angle triangle in exactly one way, but there are many more examples.

12 cm: (3,4,5)

24 cm: (6,8,10)

30 cm: (5,12,13)

36 cm: (9,12,15)

40 cm: (8,15,17)

48 cm: (12,16,20)

In contrast, some lengths of wire, like 20 cm, cannot be bent to form an integer sided right angle triangle, and other lengths allow more than one solution to be found; for example, using 120 cm it is possible to form exactly three different integer sided right angle triangles.

120 cm: (30,40,50), (20,48,52), (24,45,51)

Given that $L$ is the length of the wire, for how many values of $L \leq 1500000$ can exactly one integer sided right angle triangle be formed?

Haskell

import Data.Array

triples :: Int -> [Int]
triples limit = [a + b + c | u <- [1..l'], v <- [u+1,u+3..l'-u], gcd u v == 1,
                             let a = v^2 - u^2, let b = 2*u*v, let c = u^2 + v^2]
    where l' = round $ sqrt $ fromIntegral limit

perimeters :: Int -> Array Int Int
perimeters limit = accumArray (+) 0 (1, limit) $ map (\p -> (p, 1)) $ concat [takeWhile (<= limit) $ map (*p) [1..] | p <- triples limit]

main :: IO ()
main = print $ length $ filter (== 1) $ elems $ perimeters 1500000

$ ghc -O2 -o singular-triangles singular-triangles.hs
$ time ./singular-triangles
real   0m0.424s
user   0m0.416s
sys    0m0.008s