Tree of primitive Pythagorean triples

Presence of exclusively primitive Pythagorean triples It can be shown inductively that the tree contains primitive Pythagorean triples and nothing else by showing that starting from a primitive Pythagorean triple, such as is present at the initial node with , each generated triple is both Pythagorean and primitive. Preservation of the Pythagorean property If any of the above matrices, say , is applied to a triple having the Pythagorean property to obtain a new triple , this new triple is also Pythagorean. This can be seen by writing out each of , , and as the sum of three terms in , , and , squaring each of them, and substituting to obtain . This holds for and as well as for . Preservation of primitivity The matrices , , and are all unimodular—that is, they have only integer entries and their determinants are ±1. Thus their inverses are also unimodular and in particular have only integer entries. So if any one of them, for example , is applied to a primitive Pythagorean triple to obtain another triple , we have and hence . If any prime factor were shared by any two of (and hence all three of) , , and then by this last equation that prime would also divide each of , , and . So if , , and are in fact pairwise coprime, then , , and must be pairwise coprime as well. This holds for and as well as for . Presence of every primitive Pythagorean triple exactly once To show that the tree contains every primitive Pythagorean triple, but no more than once, it suffices to show that for any such triple there is exactly one path back through the tree to the starting node . This can be seen by applying in turn each of the unimodular inverse matrices , , and to an arbitrary primitive Pythagorean triple , noting that by the above reasoning primitivity and the Pythagorean property are retained, and noting that for any triple larger than exactly one of the inverse transition matrices yields a new triple with all positive entries (and a smaller hypotenuse). By induction, this new valid triple itself leads to exactly one smaller valid triple, and so forth. By the finiteness of the number of smaller and smaller potential hypotenuses, eventually is reached. This proves that does in fact occur in the tree, since it can be reached from by reversing the steps; and it occurs uniquely because there was only one path from to . ==Properties==

The transformation using matrix , if performed repeatedly from , preserves the feature ; matrix preserves starting from ; and matrix preserves the feature starting from . A geometric interpretation for this tree involves the excircles present at each node. The three children of any parent triangle “inherit” their inradii from the parent: the parent's excircle radii become the inradii for the next generation. For example, parent has excircle radii equal to 2, 3 and 6. These are precisely the inradii of the three children , and respectively. If either of or is applied repeatedly from any Pythagorean triple used as an initial condition, then the dynamics of any of , , and can be expressed as the dynamics of in : x_{n+3} - 3x_{n+2} + 3x_{n+1} - x_n = 0 \, which is patterned on the matrices' shared characteristic equation :\lambda ^3 -3 \lambda ^2 + 3 \lambda -1 = 0. \, If is applied repeatedly, then the dynamics of any of , , and can be expressed as the dynamics of in : x_{n+3} - 5x_{n+2} - 5x_{n+1} + x_n = 0, \, which is patterned on the characteristic equation of . Moreover, an infinitude of other third-order univariate difference equations can be found by multiplying any of the three matrices together an arbitrary number of times in an arbitrary sequence. For instance, the matrix moves one out the tree by two nodes (across, then down) in a single step; the characteristic equation of provides the pattern for the third-order dynamics of any of , , or in the non-exhaustive tree formed by . ==Alternative methods of generating the tree==

Using two parameters Another approach to the dynamics of this tree relies on the standard formula for generating all primitive Pythagorean triples: :a = m^2 - n^2, \, :b = 2mn, \, :c = m^2 + n^2, \, with and and coprime and of opposite parity (i.e., not both odd). Pairs can be iterated by pre-multiplying them (expressed as a column vector) by any of : \begin{array}{lcr} \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}, & \begin{bmatrix} 2 & 1 \\ 1 & 0 \end{bmatrix}, & \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}, \end{array} each of which preserves the inequalities, coprimeness, and opposite parity. The resulting ternary tree, starting at , contains every such pair exactly once, and when converted into triples it becomes identical to the tree described above. Alternatively, start with for the root node. Then the matrix multiplications will preserve the inequalities and coprimeness, and both and will remain odd. The corresponding primitive Pythagorean triples will have , , and . This tree will produce the same primitive Pythagorean triples, though with and switched. Using one parameter This approach relies on the standard formula for generating any primitive Pythagorean triple from a half-angle tangent. Specifically one writes , where is the tangent of half of the interior angle that is opposite to the side of length . The root node of the tree is , which is for the primitive Pythagorean triple . For any node with value , its three children are , , and . To find the primitive Pythagorean triple associated with any such value , compute and multiply all three values by the least common multiple of their denominators. (Alternatively, write as a fraction in lowest terms and use the formulas from the previous section.) A root node that instead has value will give the same tree of primitive Pythagorean triples, though with the values of and switched. ==A different tree==

Alternatively, one may also use three different matrices found by Price: : \begin{array}{lcr} A' = \begin{bmatrix} 2 & 1 & -1 \\ -2 & 2 & 2 \\ -2 & 1 & 3 \end{bmatrix}\,, & B' = \begin{bmatrix} 2 & 1 & 1 \\ 2 & -2 & 2 \\ 2 & -1 & 3 \end{bmatrix}\,, & C' = \begin{bmatrix} 2 & -1 & 1 \\ 2 & 2 & 2 \\ 2 & 1 & 3 \end{bmatrix}\,. \end{array} The three children produced by the set and the children produced by the set are not the same, but each set separately produces all primitive triples. For example, using as the parent, we get two sets of three children: : \begin{array}{ccc} & \left[ 5,12,13 \right] & \\ A & B & C \\ \left[ 45,28,53 \right] & \left[ 55,48,73 \right] & \left[ 7,24,25 \right] \end{array} \quad \quad \quad \quad \quad \quad \begin{array}{ccc} {} & \left[ 5,12,13 \right] & {} \\ A' & B' & C' \\ \left[ 9,40,41 \right] & \left[ 35,12,37\right] & \left[ 11,60,61 \right] \end{array} == Notes and references ==