Established fact
Pythagorean triples
Most right triangles have at least one side that is not a whole number. A few have three whole-number sides, and they follow patterns people have studied for almost four thousand years.
What a triple is
A Pythagorean triple is three whole numbers a, b and c with a² + b² = c², so they can be the sides of a right triangle. 3-4-5 is the smallest: 9 + 16 = 25. A triple is primitive when the three numbers share no common factor; 6-8-10 is a triple, but only a scaled-up 3-4-5.
How to make every one
Euclid gave a recipe around 300 BCE. Pick two whole numbers m > n. Then a = m² − n², b = 2mn and c = m² + n² always form a triple. When m and n share no factor and one of them is even, the triple is primitive, and every primitive triple comes from exactly one such pair.
| m, n | Triple |
|---|---|
| 2, 1 | 3 · 4 · 5 |
| 3, 2 | 5 · 12 · 13 |
| 4, 1 | 15 · 8 · 17 |
| 4, 3 | 7 · 24 · 25 |
| 5, 2 | 21 · 20 · 29 |
| 5, 4 | 9 · 40 · 41 |
The hypotenuse-plus-one family
Setting n = m − 1 gives triples whose hypotenuse is exactly one more than the longer leg: 3-4-5, 5-12-13, 7-24-25, 9-40-41, 11-60-61, and on through every odd number. The short leg is odd, and the other two are the two halves of its square: 7² = 49 = 24 + 25.
Patterns every primitive triple shares
In every primitive triple, one leg is divisible by 3, one leg is divisible by 4, and one of the three numbers is divisible by 5. The hypotenuse is always odd. It follows that the product of the two legs is always divisible by 12, so the triangle’s area is always a multiple of 6.
A 3,800-year-old table
A Babylonian clay tablet known as Plimpton 322, written around 1800 BCE, lists fifteen rows of numbers that match Pythagorean triples, some as large as 12709-13500-18541. It predates Pythagoras by over a thousand years. Historians still debate what it was for, from a teaching aid to a trigonometric table.