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The Berggren tree is a rooted tree of primitive Pythagorean triples, with three children at every node. Fix the convention $a$ odd, $b$ even and $c>0$. Starting from the column vector $(3,4,5{)}^{\mathsf{T}}$, multiply by any ofThe first children are $(5,12,13)$, $(21,20,29)$ and $(15,8,17)$. These unimodular matrices preserve the quadratic form $a^2+b^2-c^2$, integer parity and greatest common divisor. Every primitive Pythagorean triple occurs exactly once. F. J. M. Barning's report gives this matrix-generation theorem with the even leg first.^[ref]

An infinite descent argument explains completeness. Except at the root, the unique parent isIts hypotenuse is positive and smaller than $c$, since $c<a+b\le \sqrt{2}c$. The first two unmodified entries cannot both be negative; their signs select the inverse of $A$, $B$ or $C$. The odd entry cannot vanish; the even entry vanishes only at $(3,4,5)$. Repeated descent therefore reaches that root. Multiplying all three coordinates by a positive integer supplies every nonprimitive Pythagorean triple.

The formulas $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$ give a Pythagorean triple for integers $m>n>0$, because $(m^2-n^2{)}^2+(2mn{)}^2=(m^2+n^2{)}^2$. They give a primitive Pythagorean triple exactly when $m,n$ are coprime and have opposite integer parity. Every primitive Pythagorean triple arises this way after possibly exchanging the legs; scaling gives the nonprimitive triples.

F. J. M. Barning gave a matrix-based generation process for primitive Pythagorean triples. His 1963 Mathematisch Centrum report verifies the initials and presents three unimodular matrices generating all such triples without repetition, with a fixed leg-order convention.^[ref]

The transformations preserve the equation $x^2+y^2=z^2$. Starting from $(4,3,5)$, with the even leg first, repeated transformations build a branching tree of solutions. The method turns a Diophantine equation into a recursive enumeration algorithm with a uniqueness theorem.^[ref]

A Pythagorean triple $(a,b,c)$ whose coordinates have greatest common divisor 1. Its two legs have opposite integer parity, and every Pythagorean triple is a unique positive integer multiple of a primitive Pythagorean triple up to the ordering of its legs.

The equation $a^2+b^2=c^2$. Its positive integer solutions are Pythagorean triples; primitive Pythagorean triples are those with no common factor greater than 1.