Primitive Pythagorean Triples on a Curvilinear Grid Defined by Euclid's Formula

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scale
0.3
range
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A primitive Pythagorean triple is a set of values
(a,b,c)
satisfying the Pythagorean theorem
2
a
+
2
b
=
2
c
, with no common factors,
gcd(a,b,c)=1
. The triples can be rewritten using Euclid's formula as
(mn,
2
m
-
2
n
,
2
m
+
2
n
)
with
m
and
n
coprime and
m-n
odd to ensure that the triples are primitive. The primitive triples are represented by their corresponding triangles, scaled and translated to the intersection of the curves for each value
a
,
b
, in the centroid of the triangle. Empty intersections occur where the triple is not primitive.

Details

Let
m(i)=2i-1
and
n(j)=2j+1
. Each blue curve is given parametrically by
i⟶
m(i)n(j),
2
m(i)
+
2
n(j)

for fixed
j
; different values of
j
give different blue curves. The blue curves shown correspond to integer choices of
j
. Similarly, each red curve is given parametrically by
j⟶m(i)n(j),
2
n(j)
-
2
m(i)

for fixed
i
; different values of
i
give different red curves. The red curves shown correspond to integer choices of
i
.

References

[1] Wikipedia. "Pythagorean triple." (Sep 26, 2012) en.wikipedia.org/wiki/Pythagorean_triple.

External Links

Cone of Pythagorean Triplets
Plotting Pythagorean Triples
Primitive Pythagorean Triple (Wolfram MathWorld)
Primitive Pythagorean Triples 1: Scatter Plot
Primitive Pythagorean Triples 2: Ordered Pairs
Pythagorean Primitive Triples Using Primes
Pythagorean Triple (Wolfram MathWorld)

Permanent Citation

Enrique Zeleny
​
​"Primitive Pythagorean Triples on a Curvilinear Grid Defined by Euclid's Formula"​
​http://demonstrations.wolfram.com/PrimitivePythagoreanTriplesOnACurvilinearGridDefinedByEuclid/​
​Wolfram Demonstrations Project​
​Published: November 27, 2012