Tree of primitive Pythagorean triples

The tree of primitive Pythagorean triples.

In mathematics, a Pythagorean triple is a set of three positive integers a, b, and c having the property that they can be respectively the two legs and the hypotenuse of a right triangle, thus satisfying the equation $a^{2}+b^{2}=c^{2}$; the triple is said to be primitive if and only if a, b, and c share no common divisor. The set of all primitive Pythagorean triples has the structure of a rooted tree, specifically a ternary tree, in a natural way. This was first discovered by B. Berggren in 1934.

F. J. M. Barning showed that when any of the three matrices

${\begin{array}{lcr}A={\begin{bmatrix}1&-2&2\\2&-1&2\\2&-2&3\end{bmatrix}}&B={\begin{bmatrix}1&2&2\\2&1&2\\2&2&3\end{bmatrix}}&C={\begin{bmatrix}-1&2&2\\-2&1&2\\-2&2&3\end{bmatrix}}\end{array}}$

is multiplied on the right by a column vector whose components form a Pythagorean triple, then the result is another column vector whose components are a different Pythagorean triple. If the initial triple is primitive, then so is the one that results. Thus each primitive Pythagorean triple has three "children". All primitive Pythagorean triples are descended in this way from the triple (3 ,4, 5), and no primitive triple appears more than once. The result may be graphically represented as an infinite ternary tree with (3, 4, 5) at the root node (see classic tree at right). This tree also appeared in papers of A. Hall in 1970 and A. R. Kanga in 1990.

It is possible to geometrically interconnect the triples of any branch using the Central Square variant of the proof of the Pythagoras´ Theorem. The Figure below shows the example of the Pythagoras´ (upper) branch composed of a sequence of triples where the smallest side is an odd number, i.e.

$(3,4,5),(5,12,13),(7,24,25),(9,40,41)...$

The process by which this geometrical construction is done is explained below in section The Geometry of the Pythagorean Triples as presented by L. Teia in 2014.

The geometrical representation of the sequence of Pythagoras´ odd triples

Proofs

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 (3, 4, 5), each generated triple is both Pythagorean and primitive.

Preservation of the Pythagorean property

If any of the above matrices, say A, is applied to a triple (a, b, c)^T having the Pythagorean property a²+b²=c² to obtain a new triple (d, e, f)^T = A(a, b, c)^T, this new triple is also Pythagorean. This can be seen by writing out each of d, e, and f as the sum of three terms in a, b, and c, squaring each of them, and substituting c²=a²+b² to obtain f²=d²+e². This holds for B and C as well as for A.

Preservation of primitivity

The matricies A, B, and C 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 A, is applied to a primitive Pythagorean triple (a, b, c)^T to obtain another triple (d, e, f)^T, we have (d, e, f)^T = A(a, b, c)^T and hence (a, b, c)^T = A⁻¹(d, e, f)^T. If any prime factor were shared by any two of (and hence all three of) d, e, and f then by this last equation that prime would also divide each of a, b, and c. So if a, b, and c are in fact pairwise coprime, then d, e, and f must be pairwise coprime as well. This holds for B and C as well as for A.

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 (3, 4, 5). This can be seen by applying in turn each of the unimodular inverse matrices A⁻¹, B⁻¹, and C⁻¹ to an arbitrary primitive Pythagorean triple (d, e, f), noting that by the above reasoning primitivity and the Pythagorean property are retained, and noting that for any triple larger than (3, 4, 5) 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 (3, 4, 5) is reached. This proves that (d, e, f) does in fact occur in the tree, since it can be reached from (3, 4, 5) by reversing the steps; and it occurs uniquely because there was only one path from (d, e, f) to (3, 4, 5).

Properties

The transformation using matrix A, if performed repeatedly from (a, b, c) = (3, 4, 5), preserves the feature b + 1 = c; matrix B preserves a – b = ±1 starting from (3, 4, 5); and matrix C preserves the feature a + 2 = c starting from (3, 4, 5).

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 ( 3, 4, 5) has excircle radii equal to 2, 3, and 6. These are precisely the inradii of the three children (5, 12, 13), (15, 8, 17) and (21, 20, 29) respectively.

If either of A or C is applied repeatedly from any Pythagorean triple used as an initial condition, then the dynamics of any of a, b, and c can be expressed as the dynamics of x 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 B is applied repeatedly, then the dynamics of any of a, b, and c can be expressed as the dynamics of x in

$x_{n+3}-5x_{n+2}-5x_{n+1}+x_{n}=0,\,$

which is patterned on the characteristic equation of B.

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 D = CB moves one out the tree by two nodes (across, then down) in a single step; the characteristic equation of D provides the pattern for the third-order dynamics of any of a, b, or c in the non-exhaustive tree formed by D.

Alternative methods of generating the tree

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 m > n > 0 and m and n coprime and of opposite parity. Pairs (m, n) 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 contains every such (m, n) pair exactly once, and when converted into (a, b, c) triples it becomes identical to the tree described above.

Another way of using two underlying parameters to generate the tree of triples uses an alternative formula for all primitive triples:

$a=uv,\,$

$b={\frac {u^{2}-v^{2}}{2}},$

$c={\frac {u^{2}+v^{2}}{2}},$

with u > v > 0 and u and v coprime and both odd. Pairs (u, v) can be iterated by pre-multiplying them (expressed as a column vector) by any of the above 2 × 2 matrices, all three of which preserve the inequalities, coprimeness, and the odd parity of both elements. When this process is begun at (3, 1), the resulting ternary tree contains every such (u, v) pair exactly once, and when converted into (a, b, c) triples it becomes identical to the tree described above.

A different tree

A different tree found by Price may be produced in a similar way using matrices A',B',C' shown below. (See Pythagorean triples by use of matrices and linear transformations.)

${\begin{array}{lcr}A'={\begin{bmatrix}2&1&-1\\-2&2&2\\-2&1&3\end{bmatrix}}&B'={\begin{bmatrix}2&1&1\\2&-2&2\\2&-1&3\end{bmatrix}}&C'={\begin{bmatrix}2&-1&1\\2&2&2\\2&1&3\end{bmatrix}}\end{array}}$

The Geometry of the Pythagorean Triples

Under completion. Apologies for the inconvenience, and thank you for your patience.

The Pythagoras (upper) branch

The sequence of odd triples of Pythagoras are highlighted in grey at the upper most branch of the tree. As one moves along the branch, it can be observe that the minor side increases by 2 from one triple to the next starting with the square of side length 1. The relationship between major sides is less trivial to define. It can be proven that the major side of any triple is equal to the summation of its minor side with both the minor and major side of the preceding triple.

The Pythagoras´ (upper) branch of the tree of triples, and the relationship between neighbouring triples

According to the Pythagoras theorem, the hypotenuse side is then governed by the lengths of the minor and major sides. As a result, the triple at any location $n$ along the Pythagoras´ branch of odd triples is given as

$(x_{n},y_{n},z_{n})=\left\{{\begin{array}{ll}x_{n}=2n+1\\y_{n}=x_{1}+y_{1}+2\sum _{i=1}^{n-2}x_{n-i}+x_{n}\\z_{n}={\sqrt {x_{n}^{2}+y_{n}^{2}}}\\\end{array}}\right.$

This is valid for any $n\geqslant 2$. Note that the series in the term for the major side connects it with the first triple $(3,4,5)$.

The tree of squares

Each right-angled triangle revolves around a specific square. Therefore, a second tree was found by Teia superimposed to the tree of Pythagorean Triples - the tree of squares. Whilst the original tree (solid lines) interconnects triangles of integer sides, the second tree (dashed lines) interconnects squares of interger sides about which the triangles revolve. For example,

   * triangles (3, 4, 5) and (20, 21, 29) revolve around a square of side 1
   * triangles (5, 12, 13) and (15, 8, 17) revolve around a square of side 7
   * as an extreme, triangle (105, 208, 233) revolves around a square of side 103

Tree of squares superimposed to the Tree of Primitive Pythagorean Triples

External links

• The Trinary Tree(s) underlying Primitive Pythagorean Triples at cut-the-knot
• Frank R. Bernhart, and H. Lee Price, "Pythagoras' garden, revisited", Australian Senior Mathematics Journal 01/2012; 26(1):29-40.[1]