How many primitive Pythagorean triples have hypotenuse less than 50? A triple is primitive when the three numbers share no common factor greater than 1.

Seven primitive triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (20,21,29), (12,35,37) and (9,40,41).

Every primitive triple comes from a pair of coprime integers m > n of opposite parity, via a = m^2 - n^2, b = 2mn, c = m^2 + n^2.

Primitive Triangles With a Short Hypotenuse

Issue 28 Core Published 16 Oct 2026 By The Chalkboard Desk

The problem

How many primitive Pythagorean triples have hypotenuse less than ? A triple is primitive when the three numbers share no common factor greater than .

Take a minute

Reveal hint 1

Every primitive triple comes from a pair of coprime integers of opposite parity, via , , .

Reveal hint 2

The hypotenuse is , so the constraint bounds by .

AnswerSeven primitive triples: , , , , , and .

Derivation