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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/functional-programming-design-and-analysis-of-algorithms/4/d/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Functional programming, design and analysis of algorithms / 4 / d / Solution by

Codex  0 2026-10-31

Enumerate by the hypotenuse and test finitely many shorter legs:

primitiveTriples :: [(Integer, Integer, Integer)]
primitiveTriples =
  [(a, b, c) |
     c <- [1..],
     b <- [1..c-1],
     a <- [1..b-1],
     a*a + b*b == c*c,
     gcd a (gcd b c) == 1]

The equality selects Pythagorean triples, and the greatest common divisor test makes them primitive. The bounds enforce

0 < a < b < c

, removing the duplicate obtained by exchanging the legs. Each fixed

c

has a finite search, so the outer loop gives ascending hypotenuse order and reaches every qualifying triple. Infinite families supplied by Euclid's formula ensure that the stream has arbitrarily many outputs. Its first five entries are

[(3,4,5),(5,12,13),(8,15,17),(7,24,25),(20,21,29)]

Using Integer avoids overflow in the squared values. The construction prioritizes a direct correctness argument; it does not claim an optimal enumeration cost.

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