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

In summary, Tony has defined "Generalized Pell Numbers" in his document and is interested in knowing if they are already known and if they have any other properties. He is particularly interested in their period modulo a prime number and is hoping that these numbers will provide insight for a primality test for Fermat numbers. He has added more information in the paper and has found that these numbers share interesting properties modulo a prime number that could potentially lead to a primality test.
  • #1
T.Rex
62
0
Hi,
In the following document: Generalized Pell Numbers, I've defined what I call "Generalized Pell Numbers".
They provide a way for computing: [tex]1+\sqrt[m]{m}[/tex].

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
 
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  • #2
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
 
  • #3


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.
 

1. What are Generalized Pell Numbers?

Generalized Pell Numbers are a sequence of integers that are similar to the well-known Pell Numbers, but with a different starting point and recurrence relation. They are defined as Pn = aPn-1 + bPn-2, where P0 = 0 and P1 = 1. The values of a and b can vary, leading to different sequences of numbers.

2. What is the significance of Generalized Pell Numbers?

Generalized Pell Numbers have applications in various fields such as number theory, combinatorics, and computer science. They also have connections to other important sequences like the Fibonacci numbers and Lucas numbers. In addition, they can be used to generate interesting geometric patterns and shapes.

3. How are Generalized Pell Numbers different from Pell Numbers?

The main difference between Pell Numbers and Generalized Pell Numbers is their starting point and recurrence relation. Pell Numbers are defined as Pn = 2Pn-1 + Pn-2, where P0 = 0 and P1 = 1. Generalized Pell Numbers have a different starting point and can have different values for a and b in the recurrence relation.

4. What is the formula for calculating Generalized Pell Numbers?

The formula for calculating Generalized Pell Numbers is Pn = aPn-1 + bPn-2, where P0 = 0 and P1 = 1. The values of a and b can be chosen based on the desired sequence of numbers. For example, if a = 2 and b = 1, the resulting sequence is the Pell Numbers.

5. Are there any interesting properties of Generalized Pell Numbers?

Yes, there are several interesting properties of Generalized Pell Numbers. They have a periodicity of 2 or 4, depending on the values of a and b. They also have a connection to the golden ratio and can be used to approximate irrational numbers. Additionally, they exhibit self-similarity and can be used to generate fractal patterns.

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