Primitive Pythagorean Triples 2: Ordered Pairs

​
given PPT
1
{ x, y, z } =
change signs
-x
-y
-z
{3,4,5}
then add n
0
to get
parent
Multiplying all three sides of a right triangle by the same number yields another right triangle because of similarity. This Demonstration illustrates that it is always possible to find a number
n
to add to all three elements of any Pythagorean triple to generate another Pythagorean triple.

Details

Let
{x,y,z}
be a Pythagorean triple (PT), that is,
x
,
y
, and
z
are positive integers such that
2
x
+
2
y
=
2
z
. A primitive Pythagorean triple (PPT) is a PT with
gcd(x,y,z)=1
.
If negative values for
{x,y,z}
are allowed, then a second PT can be obtained by adding a single unique integer value
n
to each of the three original PT elements:
{x+n,y+n,z+n}
. Redundant forms of PTs are obtained by allowing negative values. The unique value of
n
that works is
n=-2(x+y-z)
.
The triangle
T'
represented by the derived PT has the following properties.
T'
is also a right triangle.
T'
is not similar to the given triangle.
T'
is represented by a PPT in redundant form when the original triangle represents a PPT.
This pairing of PPTs provides a way to number them.
Notice that names have been given to the results. The identification of parent and the numbering of children is used to make the set of all PPTs a well-ordered set. This ordering was used to implement the "given PPT" slider (with values shown in base 3).
The simple quadratic form of the graph displayed shows that
n
exists and is unique.
Assume that
2
(z+n)
-
2
(x+n)
-
2
(y+n)
=0
and
2
z
-
2
x
-
2
y
=0
,
n≠0
.
Then
2
(z+n)
-
2
(x+n)
-
2
(y+n)
-
2
z
-
2
x
-
2
y
=0
.
Expand and cancel to get
-n(n+2x+2y-2z)=0
,
so that the unique solution is
n=-2(x+y-z)
.

External Links

Pythagorean Theorem (Wolfram MathWorld)
Pythagorean Triple (Wolfram MathWorld)
Primitive Pythagorean Triple (Wolfram MathWorld)
Pythagorean Triangle (Wolfram MathWorld)
Primitive Pythagorean Triples 1: Scatter Plot
Primitive Pythagorean Triples 3: Ordered Tree Graph
Primitive Pythagorean Triples 4: Ordered Tree Matrices

Permanent Citation

Robert L. Brown
​
​"Primitive Pythagorean Triples 2: Ordered Pairs"​
​http://demonstrations.wolfram.com/PrimitivePythagoreanTriples2OrderedPairs/​
​Wolfram Demonstrations Project​
​Published: April 11, 2011