A triple of positive integers, with , is called pythagorean triple; that is the lengths of the sides of pythagorean triangles, if and only if

.

A pythagorean triangles is primitive if and are relatively prime, so that :

There is a method of finding primitive pythagorean triangles using Gaussian integers . [2]

Let where , and are pairwise, relatively prime, positive integers. Let and , where . Then and are Gaussian Integers, and is the conjugate of . Note that

and

Since then . Hence each of and is a square. That is, there exists integers and such that and . So,

Since is a positive integer, and must be positive integers, . And since , and must be relatively prime and of opposite parity. We have the identity,

The absolute values are necessary since the terms on the left, depending on , may, or may not, be positive.

When the hypotenuse is to a power of the form , where is a non-negative integers, then there exists positive integers and such that :

and

Note that, if then interchange the labels. Clearly, implies . The parameters and have opposite parity. Therefore . Hence and have opposite parity.

Thus, all primitive Pythagorean triples of the form are given by the parametric equations :

and

Then

Barnes[1] showed that if is a solution to a primitive Pythagorean triangle, where is a non-negative integer, then every Mersenne prime less than or equal to divides . This imples that the area of the primitive pythagorean triangle is a multiple of the perfect number.

Let be a positive integer such that is a prime (Mersenne prime). And let be a primitive Pythagorean triangle. Then, the area is a multiple of the perfect number

Theorem 1Let be a primitive Pythagorean triangle where . Let be a positive integer divisor of . Let . If is a prime then .

**Proof**

Since , then

for some .

If is a prime, then since ,

.

Which implies

.

Similarly,

.

Which implies

divides .

Therefore .

Corollary 1Let be a primitive Pythagorean triangle where is a nonnegative integer. Let be any Mersenne prime less than or equal to . Then

divides .

**Proof**

Let in theorem (1). Then

From Theorem (1), every prime in the set divides .

Theorem 2If

is a primitive Pythagorean triangle where is a nonnegative integer then divides .

**Proof**

By induction on : If then . So there exists integers and , one odd the other even, such that and . Hence divides . Assume true for , then

Let and . Then if divides , divides .

**Perfect numbers and the area of primitive pythagorean triangles** :

From corollary and theorem 2 we see that the area of the triangle, the smallest Pythagorean triangle, is , the smallest perfect number.

Let be a positive integer such that is a prime (Mersenne prime). And let be a primitive Pythagorean triangle. Then, the area is a multiple of the *perfect number*

**Examples**

: Two primitive Pythagorean triangles and . Since is a prime, we know that each area must be a multiple of the perfect number . And indeed, and .

: Two primitive Pythagorean triangles and . Since is a prime, we know that each area must be a multiple of the perfect number . And indeed, and .

: Two primitive Pythagorean triangles and . Since is a prime, we know that each area must be a multiple of the perfect number . And indeed, and .

**Note:** Since , is also a multiple of each of 6 and 28.

That is,

- If is a primitive Pythagorean triangle then its area is a multiple of the perfect number .
- If is a primitive Pythagorean triangle then its area is a multiple of each of the perfect numbers and
- If is a primitive Pythagorean triangle then its area is a multiple of each of the perfect numbers , and
- If is a primitive Pythagorean triangle then its area is a multiple of each of the perfect numbers , and .
- If is a primitive Pythagorean triangle then its area is a multiple of each of the perfect numbers , and
- And, in general, if is a primitive Pythagorean triangle where is a Mersenne prime then the triangle’s area is a multiple of each perfect number less than or equal to the perfect number .

**References :**

[1] Barnes, Fred. *Primitive Pythagorean triangles where the hypotenuse is to a power*, available at pythag.net/node6.html

[2] Conrad, Keith. *The Gaussian Integers*, available at math.uconn.edu/~kconrad/blurbs/ugradnumthy/Zinotes.pdf

Very interesting. I noticed they found a new Mersenne prime today, started wondering if there was a way to predict Mersenne primes based on Pythagorean triples. I see you have already been thinking along these lines. Thanks.