# Help about properties of Pell numbers

• T.Rex
In summary, the conversation is about properties of Pell numbers and the speaker asks for help on various proofs related to these numbers. They have already found a proof for C1 and mention a mistake in C4. They then ask for help and mention Fermat prime numbers in their proposed solution.
T.Rex
Hi, I need some help about properties of Pell numbers:

$$U_n = 2 U_{n-1} + U_{n-2} , \text{ with: } U_0=0 \text{ and: } U_1=1$$

$$V_n = 2 V_{n-1} + V_{n-2} , \text{ with: } V_0=2 \text{ and: } V_1=2$$

I have a proof for:

$$\frac{V_{\displaystyle 2^{\scriptstyle n}}}{2} \ = \ 1 + 4 \prod_{i=0}^{n-2}V_{\displaystyle 2^{\scriptstyle i}}^{\scriptstyle 2} \ \equiv \ 1 \pmod{2^{\scriptstyle 2n}} \ \text{\ \ \ \ (for } n \geq 2 )$$

But I have no proof for C1:

$$\frac{V_{\displaystyle 2^{\scriptstyle n}+1}}{2} \ \equiv \ 1 \pmod{2^{\scriptstyle n+1}} \ \text{\ \ \ \ (for } n \geq 2 )$$

and C2:

$$p,q \ \text{ odd primes }, \ \ p \mid V_{\displaystyle q} \ \ \Longrightarrow \ \ p \equiv 1 \pmod{2q}$$

and C3:

$$p \ \text{ odd prime }, \ \ p \mid V_{\displaystyle 2^{\scriptstyle i}} \ \ \Longrightarrow \ \ p \equiv 1 \pmod{2^{\scriptstyle i+2}}$$

and C4 (a guess):
$$p \ \text{ odd prime }, \ \ p \mid V_{\displaystyle 2^{\scriptstyle i}} \ \text{ and } \ p = 1 + {(2^{\scriptstyle i}\alpha)}^2 \ \ \Longrightarrow \ \ \alpha = \prod_{j=0}^{k} F_j \ \text{ or } \ \alpha = 1$$.

Can you help ?
Thanks,

Tony

Any idea ?

OK, I've got a proof for C1. It was not so difficult.

About C4, there was a mistake:
$$p \ \text{ odd prime }, \ \ p \mid V_{\displaystyle 2^{\scriptstyle n}} \ \text{ and } \ p = 1 + {(2^{\scriptstyle n}\alpha)}^2 \ \ \Longrightarrow \ \ \alpha = \prod_{j} F_j$$
where $$F_j$$ are Fermat prime numbers.

Any idea ?
tony

Hi Tony,

I would be happy to help you with your questions about properties of Pell numbers. Firstly, let's review some basic properties of Pell numbers:

1. The Pell numbers are a sequence of numbers that start with 0 and 1, and each subsequent number is the sum of twice the previous number and the number before it. This can be represented by the formula U_n = 2U_{n-1} + U_{n-2}.

2. Similarly, there is a second sequence of numbers called the Companion Pell numbers that start with 2 and 2, and each subsequent number is the sum of twice the previous number and the number before it. This can be represented by the formula V_n = 2V_{n-1} + V_{n-2}.

3. The Pell numbers and Companion Pell numbers have a special relationship, where V_n = U_{n+1}. This means that the nth Pell number is equal to the (n+1)th Companion Pell number.

C1: This statement is actually a special case of the general formula you provided in your question. If we let n = 1, then we have V_{2} = 2V_{1-1} + V_{1-2} = 2V_0 + V_{-1}. Since V_0 = 2 and V_{-1} = 2, we can plug in these values to get V_2 = 2(2) + 2 = 6. Dividing this by 2, we get 3, which is equivalent to 1 (mod 2^2). This is the same as saying V_{2+1}/2 \equiv 1 (mod 2^{2+1}).

C2: To prove this statement, we can use the fact that V_n = U_{n+1}. Let's assume that p is an odd prime and p divides V_q. This means that p divides U_{q+1}. Using the formula for Pell numbers, we can see that U_{q+1} = 2U_q + U_{q-1}. Since p divides both U_q and U_{q-1}, it must also divide U_{q+1}. This means that p divides U_{q+2}, and so on. Eventually, we will reach a point where p divides

## 1. What are Pell numbers?

Pell numbers are a sequence of integers that start with 0 and 1, and each subsequent number is the sum of twice the previous number and the number before that. The sequence goes like this: 0, 1, 2, 5, 12, 29, 70, 169, etc.

## 2. What is the significance of Pell numbers in mathematics?

Pell numbers have several important mathematical properties, including their relationship to the golden ratio and to the solutions of certain mathematical equations. They also have applications in number theory, combinatorics, and geometry.

## 3. How are Pell numbers related to the Fibonacci sequence?

Pell numbers are closely related to the Fibonacci sequence, as they both follow a similar pattern of adding the two previous numbers to get the next number. However, the Fibonacci sequence starts with 0 and 1, while Pell numbers start with 0 and 1 and then add the previous number twice instead of just once.

## 4. Are there any practical applications of Pell numbers?

While Pell numbers may not have direct practical applications, their properties and relationships have been used in fields such as cryptography, coding theory, and computer science.

## 5. How can I calculate Pell numbers?

There are several methods for calculating Pell numbers, including using a recursive formula or a matrix method. There are also online calculators and computer programs available to quickly generate a sequence of Pell numbers.

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