a Pythagorean triple
A Pythagorean triple is a set of three whole numbers that can be the sides of a right triangle — three positive integers a, b, c with a^2 + b^2 = c^2. The most famous is (3, 4, 5): 9 + 16 = 25. Because the sides come out as exact whole numbers, these triples are prized in classroom problems and in any setting where you want a clean right angle without messy square roots.
Multiplying every member of a triple by the same whole number gives another triple — (3, 4, 5) scales up to (6, 8, 10), (9, 12, 15), and so on. A triple whose three numbers share no common factor greater than 1 is called primitive; (3, 4, 5) and (5, 12, 13) are primitive, while (6, 8, 10) is not. Every primitive triple can be generated by Euclid's formula: pick two coprime positive integers m greater than n of opposite parity, and set a = m^2 - n^2, b = 2mn, c = m^2 + n^2. For m = 2, n = 1 this gives (3, 4, 5); for m = 3, n = 2 it gives (5, 12, 13). The formula produces every primitive triple exactly once, so there are infinitely many.
Triples are a delightful meeting point of geometry and number theory. They let you recognise right triangles at a glance and build them with whole-number measurements. A useful honesty: a Pythagorean triple is just an integer solution of the same equation a^2 + b^2 = c^2 — having one does not make a triangle 'more right' than any other right triangle; plenty of perfectly good right triangles (like the 1-1-sqrt(2)) have irrational sides and are not triples at all.
Euclid's formula with m = 4, n = 1 gives a = 16 - 1 = 15, b = 2·4·1 = 8, c = 16 + 1 = 17. Check: 15^2 + 8^2 = 225 + 64 = 289 = 17^2. So (8, 15, 17) is a primitive triple.
Euclid's formula a = m^2 - n^2, b = 2mn, c = m^2 + n^2 churns out every primitive triple.
Memorising a few triples — (3,4,5), (5,12,13), (8,15,17), (7,24,25) — speeds up problems, but do not assume a right-triangle side question must have an integer answer. Most real right triangles have irrational sides and are not triples.