Pythagorean Primitive Triples Using Primes

​
p
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
include p as an additional factor of C once and ...
A twice
B twice
both A and B
enforce rules:
flags PPT rule violations
undo
reset to {3, 4, 5}
reset to {4, 3, 5}
A = 1 =
1
1
B = 2 =
1
2
C = 2 =
1
2
{x, y, z} = {A + C, B + C, A + B + C} =
​{
3
,
4
,
5
}
Primitive Pythagorean triangles (PPTs) are right triangles that have integer sides with no common divisor. They were used by the Babylonians for use in excavation and surveying. Methods for finding the triples that represent PPTs were valued. A novel method is shown here.
Any (and all) Pythagorean triples
{x,y,z}
can be derived starting from two integers
A
and
B
. You can change the values of
A
and
B
by multiplying their current values by prime numbers. To do this, first click one of the two reset buttons and then repeat the following two steps:
1) choose a prime value from the setter bar at the top.
2) click one of the buttons labeled "A twice", "B twice", or "both A and B".
To generate PPTs, keep
A
and
B
relatively prime as described in the details.

Details

The early Greeks used Euclid's formula to calculate PPTs from two relatively prime integers
u
and
v
, where
v>u
:
{x,y,z}={
2
v
-
2
u
,2uv,
2
u
+
2
v
}
.
The parametrization offered here is similar and amounts to a change of variables. Let
A
and
B
be relatively prime and
2
C
=2AB
. Then
{A+C,B+C,A+B+C}
is a PPT.
Not only is this parametrization very pretty, it can be generalized and has been used in the examination of Fermat's last theorem.
In order to generate only PPTs,
A
and
B
must be relatively prime.
1) Never press the "both A and B" button.
2) Never include a prime as an additional factor in
A
or
B
if that prime is already a factor of the other integer.
Select the checkbox "enforce rules:" to be warned that the relatively prime rule is violated.
The prime factors of the resulting PPT sides can be examined by mousing over the number of interest.

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

Permanent Citation

Robert L. Brown
​
​"Pythagorean Primitive Triples Using Primes"​
​http://demonstrations.wolfram.com/PythagoreanPrimitiveTriplesUsingPrimes/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011