There are infinitely many primitive Pythagorean triples , like ,

with hypotenuse c such that the odd leg is a pentagonalnumber and the even leg is consecutive with the hypotenuse.

The odd leg a being the pentagonal number

The even leg

The hypotenuse

The odd pentagonal numbers can be partitioned into two sets by

with

with

If , then

Since the equation has a positive integer solution for

there are infinitely many Pythagorean triples of the desired form.

Here are the first few Primitive triples:

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……….

……….

……..

……..

…….

…….

…..

…..

…..

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If , then

there are infinitely many Pythagorean triples of the desired form.

Here are the first few Primitive triples:

……………

…………

……….

……….

……..

……..

…….

…..

…..

…..

…..

………………

………………