Finding parametric equations for 120 and 60 degree triples

Download *Deriving 120 degree and 60 degree triples*.
triples10.pdf

- means divides .
- means does not divide .
- means ``implies''.
- If and are integers then means that is their greatest common divisor. If then and are relatively prime.
- In this paper the equation for a triangle will be called a triangle. That is, will be called a 120 degree triangle. And I will write ``the 120 degree triangle ''.

A 120 degree triple is a solution in positive integers to the equation

If, additionally, , and are pairwise relatively prime then is a primitive 120 degree triple, and is a primitive 120 degree triangle. Clearly, if some integer divides any two of the variables , , and then it divides the remaining variable, therefore can be divided out. Similarly, if is primitive then multiply through by the appropriate integer to get your chosen non-primitive solution. Hence, to find all solutions, it's sufficient to find all primitive solutions.

We will need a preliminary result.

, . So, and and

. Let where and are positive integers.

- If where or 2, then . This implies that or 1. Therefore, if then , and if then .

, and , such that

and

Let be a primitive 120 degree triangle. From the triangle inequality, and since and are each non-zero, the sum of the lengths of any two sides is greater than the length of the remaining side. That is, . Therefore, there exists relatively prime positive integers and , , such that

Thus

reduced to lowest terms. From whence

Square both sides of (29), multiply through by , then cancel and rearrange terms to get the result,

There are two cases, and .

**Case I,
**. From Lemma (1),
. Also
. Thus, from
(30),

and from (29), | (31) |

**Case II,
**. From Lemma
(1),
.
Then, from (30),

To show that (32) generates the same solutions as (31), note that and . So let

and

Then

and

Substituting these values for and into (32) yields

where and are relatively prime, positive integers, and . Hence (33), with the labels for and interchanged, is identical to (31). And in the other direction of the if and only if, indeed

Therefore, all 120 degree triples where is considered the same as are given by the parametric equations,

where and are relatively prime, positive integers, , , and is some positive integer.

if and only if there exists relatively prime, positive integers and , , such that

and

for some positive integer ..

Let

where , and are pairwise, relatively prime, positive integers. Set , and let and where

and

Then and are Eisenstein Integers, and is the conjugate of . Note that , and . Thus

Since then . Hence each of and is a square. That is, there exists integers and such that and . So, from equation (36),

and |

Since is a positive integer, and must be positive integers, . And since then and must be relatively prime, and .

If , and are positive integers such that then is a 60 degree triple. If additionally ,and are pairwise relatively prime then is a primitive 60 degree triple. If is a primitive 60 degree triple then is a primitive 60 degree triangle, and vice versa.

Parametric equations for finding all 60 degree triples can be easily derived from the parametric equations for finding all 120 degree triples. (34)

First note that for any positive integer , is a 60 degree triple. That is, where is the length of each side of an equilateral triangle. Clearly, if is a 60 degree triple where then . So, since we already know how to write down the side lengths of an equilateral triangle, the triples such that need not be included in our derivation.

- [Case I,
] We have
.
Hence
is a
triple. Set
,
,
and
. Then
,
, and
.
And in the other direction, .

- [Case II,
] We have
. Hence
is a
triple.
Set
, b=
, and
. Then
,
, and
.
And in the other direction,

From equation (34) and Claim (7), where and , if , is a 60 degree triple if and only if there exists relatively prime positive integers, and , , and such that

and

or

and

Where is some positive integer.

Figure (1) illustrates Claim (7) where in and , and in and .