Generalized Fibonacci and Lucas Numbers.

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  • #1
meow91006
1
0
[h=2][/h]
Can you help me prove this theorem regarding Fibonacci and Lucas numbers?

Theorem.

Let m,r ϵ Z and n be non-zero integer. Then

U2mn+r ≡ (-1)mn Ur (mod Um) and

V2mn+r ≡ (-1)mn Vr (mod Um).Im not that good at proving. This type of congruence is much harder than what I read in our book, but I badly need the proof for this one, even just this one, to understand better Fiboancci and Lucas numbers.

I'd be glad to hear from you soon.
Thank you very much!​
 
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  • #2
meow91006 said:
Can you help me prove this theorem regarding Fibonacci and Lucas numbers?

Theorem.

Let m,r ϵ Z and n be non-zero integer. Then

U2mn+r ≡ (-1)mn Ur (mod Um) and

V2mn+r ≡ (-1)mn Vr (mod Um).Im not that good at proving. This type of congruence is much harder than what I read in our book, but I badly need the proof for this one, even just this one, to understand better Fiboancci and Lucas numbers.

I'd be glad to hear from you soon.
Thank you very much!​

That is not a question as it stands, please post the full question.

CB
 

FAQ: Generalized Fibonacci and Lucas Numbers.

What are Generalized Fibonacci and Lucas Numbers?

Generalized Fibonacci and Lucas Numbers are sequences of numbers that follow a specific mathematical pattern. These sequences are similar to the original Fibonacci and Lucas sequences, but they use different starting values and different recurrence relations. They have been studied extensively in mathematics and have applications in various fields such as computer science, biology, and economics.

What is the formula for Generalized Fibonacci and Lucas Numbers?

The formula for Generalized Fibonacci and Lucas Numbers is Fn = aFn-1 + bFn-2, where Fn represents the nth term in the sequence, and a and b are the starting values. The formula can be modified for different starting values and can also include higher order terms.

What are the applications of Generalized Fibonacci and Lucas Numbers?

Generalized Fibonacci and Lucas Numbers have various applications in mathematics and other fields. They can be used to model population growth, analyze the efficiency of algorithms, and study the growth patterns of plants and animals. They also have applications in coding theory, number theory, and financial markets.

How do Generalized Fibonacci and Lucas Numbers differ from the original Fibonacci and Lucas sequences?

The main difference between Generalized Fibonacci and Lucas Numbers and the original sequences is their starting values and recurrence relations. The original sequences use starting values of 0 and 1, while the generalized versions can have any starting values. Additionally, the recurrence relation for the original sequences is Fn = Fn-1 + Fn-2, while the generalized versions can have different recurrence relations depending on the starting values.

How can Generalized Fibonacci and Lucas Numbers be extended to higher dimensions?

Generalized Fibonacci and Lucas Numbers can be extended to higher dimensions by using matrices. The recurrence relation for the generalized sequences can be rewritten in matrix form, and the resulting matrix can be raised to higher powers to generate terms in the sequence. This technique is useful in analyzing complex systems and can also be applied in coding theory and cryptography.

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