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Concept

Primitive triple

A Pythagorean triple whose three numbers share no common factor, so that it is not a whole multiple of a smaller one. Every triple is a multiple of exactly one primitive one, and the primitive ones are what a parametrisation or a tree is asked to produce.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

(−1, 0)3/4 → 7, 24, 252/3 → 5, 12, 131/2 → 3, 4, 52/5 → 21, 20, 291/3 → 8, 6, 101/4 → 15, 8, 17each line of rational slope meets the circle a second time at a rational pointclearing the denominators turns that point into a Pythagorean triple, and every triple arises this way

Every triple, on one circle

Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.

number · pythagoras 3-4-55-12-137-24-2548-55-7328-45-5320-21-2939-80-89119-120-16936-77-858-15-1733-56-6565-72-9712-35-37the rootfirst matrixsecondthird13 triples in 3 generations, each child got from its parent by one of three fixed matrices; the root is the 3-4-5 trianglegrown until the hypotenuse passes 200, the tree produces exactly the 32 primitive triples an exhaustive search over Euclid's m and n finds— the same set, by two procedures with nothing in common

A tree that holds every triple

Three fixed matrices, applied to 3-4-5 over and over, produce every primitive Pythagorean triple there is — each of them once, none of them twice, and with no test for common factors anywhere in the procedure.

number · pythagoras

Named alongside it

The objects these essays reach for when they reach for this one.

BijectionDescentPythagorean triplesConicHypotenuseMatrixParametrisationProjectionRational pointsRecurrenceRight triangleStereographic projection

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