Fibonacci or Lucas Number
Date: 02/19/2003 at 17:19:47 From: Alondra Subject: Fibonacci or Lucas number How do I know that any number x is a Fibonacci or Lucas number? For example, how can I know that 14930352 is a Fibonacci number without doing the whole sequence? Is there any way, just like I know that 113 is a prime number? Has anyone found that way? Also, my Math Fair group really wants to do a project about the relation between Fibonacci and Lucas numbers. We've placed both sequences on the Spirals of Ulam, and many other different things, and we even found a way to find the constant in both numbers using a formula, without having to complete the instructions of the Ulam, but we get stuck there. I was wondering if there's an incomplete work we could continue, or something like that. Thank you.
Date: 02/20/2003 at 03:50:48 From: Doctor Floor Subject: Re: Fibonacci or Lucas number Hi, Alondra, Thanks for your question. In the Dr. Math archives you can find direct formulas for the Fibonacci and Lucas sequences: A Fibonacci and Lucas Number Relation http://mathforum.org/library/drmath/view/52699.html With P = (1+SQRT(5))/2 (a.k.a. the Golden Ratio) Q = (1-SQRT(5))/2 these formulas are F(n) = (P^n - Q^n)/SQRT(5) L(n) = P^n + Q^n Now the absolute value |Q^n|<0.5 for n>1. This means that for n>2 you can in both formulas remove Q^n and use rounding instead: F(n) = round(P^n/SQRT(5)) L(n) = round(P^n) With this, it is easy to find a possible value of n using logarithms. Let's take your example: you want to know whether there is an n for which F(n)=14930352. We find that log(sqrt(5)*14930352)/log(P) is approximately equal to 36. So 14930352 might be F(36). We check by calculating round(P^36/sqrt(5)) and indeed this is equal to 14930352. I think this is the quickest way of checking. If you have more questions, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/
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