Primitive Pythagorean Triples 4: Ordered Tree Matrices

given PPT

ID number

0

is

{a, b, c}

matrix

P

C

i

C

i

derivation

parent	{ x p , y p , z p } =	{a,b,c} ·	P
child 0	{ x 0 , y 0 , z 0 } =	{-a,-b,c} ·	P
child 1	{ x 1 , y 1 , z 1 } =	{a,-b,c} ·	P
child 2	{ x 2 , y 2 , z 2 } =	{-a,b,c} ·	P

Given a primitive Pythagorean triple (PPT), three new PPTs, which we call children of the original, are generated by multiplying the original on the right by three fixed matrices called Hall matrices. Moreover, any PPT can be generated by a series of such right-multiplications starting from the PPT

{3,4,5}

. Equivalently, we can generate the three new PPTs by right-multiplying "redundant forms" of the original PPT by a single matrix. The PPT

{3,4,5}

has a total of eight redundant forms,

{±3,±4,±5}

, but only the four redundant forms,

{±3,±4,+5}

, are required.

In this Demonstration, the single matrix is denoted

P

. The "

C

i

derivation" button shows how the Hall matrices are computed as a combination of

P

and other matrices described in Details, and the "

C

i

"

button shows the generation of children using Hall matrices.