Are Generalized Pell Numbers Already Known and What are Their Properties?

[T.Rex]

Hi,
In the following document: Generalized Pell Numbers, I've defined what I call "Generalized Pell Numbers".
They provide a way for computing: $$1+\sqrt[m]{m}$$.

I'd like to know if these numbers are already known or not, and if someone knows about other properties they have or if someone is interested to look after new properties.

My main goal is to get more information about the period of Pell Numbers modulo a prime number. So my hope is that these Generalized Pell Numbers will lead to something that could help me.

If you know about properties of Pell numbers modulo a prime number, just let me know !

Tony

 

[T.Rex]

More interesting info: primality test for Fermat numbers ?

Hi,

I've added more information in the paper. A preliminary study seems to show that all these numbers share interesting properties modulo a prime number (including a Fermat prime number) that could lead to a primality test, once proofs are provided ...

Tony

 

[M98Ranger]

Hello Tony,

Thank you for sharing your work on Generalized Pell Numbers. To answer your question, yes, these numbers have been studied before and have been found to have interesting properties. In fact, they are a generalization of Pell numbers, which have been studied extensively in number theory.

Some other properties of Generalized Pell Numbers include their relationship to Lucas sequences and their connection to the golden ratio. They also have applications in algebraic number theory and have been studied for their divisibility properties.

As for your main goal, there is ongoing research on the period of Pell Numbers modulo a prime number. Your work on Generalized Pell Numbers may indeed contribute to this area of study and could potentially lead to new insights. I would recommend reaching out to other researchers or joining a number theory community to discuss your findings and potentially collaborate on further investigations.

I wish you all the best in your research and hope you continue to explore the fascinating world of Generalized Pell Numbers.