Generating sorted list of all possible coprimes

Question

I need to generate infinite sorted list of all coprimes. The first element in each pair must be less than the second. The sorting must be done in ascending order -- by the sum o

Answer 1

Here's a possible solution following Chapter 9 of Richard Bird's Thinking Functionally in Haskell:

coprimes = mergeAll $ map coprimes' [2..]

coprimes' n = [(n, m) | m <- [n+1..], gcd m n == 1]

merge (x:xs) (y:ys)
    | s x < s y =  x:merge xs (y:ys)
    | s x == s y = x:y:merge xs ys
    | otherwise = y:merge (x:xs) ys
    where s (x, y) = x+y

xmerge (x:xs) ys = x:merge xs ys

mergeAll = foldr1 xmerge

And the result is:

> take 10 $ coprimes
[(2,3),(2,5),(3,4),(3,5),(2,7),(4,5),(3,7),(2,9),(3,8),(4,7)]

Note that the natural definition of mergeAll would be foldr1 merge, but this doesn't work because it will try to find the minimum of the first elements of all the list before returning the first element, and hence you end up in an infinite loop. However, since we know that the lists are in ascending order and the minimum is the first element of the first list xmerge does the trick.

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Note: this solution appears to be significantly (like 2 order of magnitudes) slower than Carsten "naive" answer. So I advise to avoid this if you are interested in performance. Yet it still is an interesting approach that might be effective in other situations.