Twin and Nearly Twin Pythagorean Triples

n

3

Definitions:

T

1

= (3, 4, 5)

T

2

= (4, 3, 5)

C

0

=

1	2	2
2	1	2
2	2	3

C

1

=

-1	-2	-2
2	1	2
2	2	3

C

2

=

1	2	2
-2	-1	-2
2	2	3

formula	ID number	Pythagorean triple
Twin Leg-Leg
T 1 · 3 C 0	1000	{697,696,985}
T 2 · 3 C 0	2000	{696,697,985}
Nearly Twin Leg-Hypotenuse
T 1 · 3 C 1	1111	{63,16,65}
T 2 · 3 C 2	2222	{16,63,65}
Twin Leg-Hypotenuse
T 2 · 3 C 1	2111	{40,9,41}
T 1 · 3 C 2	1222	{9,40,41}

A Pythagorean triple

{x,y,z}

is an ordered set of three positive integers that are the side lengths of a Pythagorean triangle, so that

2

x

+

2

y

=

2

z

. A twin Pythagorean triple is a Pythagorean triple in which two elements differ by one. In a twin leg-leg Pythagorean triple,

x

and

y

differ by 1; in a twin leg-hypotenuse

z-x=1

or

z-y=1

; in a nearly twin leg-hypotenuse triple,

z-x=2

or

z-y=2

.

The twin triples

T

1

={3,4,5}

and

T

2

={4,3,5}

are called the base twin triples. This Demonstration shows that all twin Pythagorean triples can be calculated by a base twin repeatedly multiplied on the right

n

times by a carefully chosen matrix.