In May 2017, I posted my first research regarding to the relationship between prime number and Fibonacci number https://www.linkedin.com/pulse/relationships-between-prime-number-fibonacci-thinh-nghiem/ I have chance to go further in this subject. In detail, I realized that a prime number can be analyzed into sum of many Fibonacci numbers. Below are some examples: 29 = 21 + 3 + 5 107 = 89 + 13 + 5 1223 = 987 + 233 + 3 I have successfully analyzed the first 1,000 prime numbers with above methodology. Calculation can be found in https://docs.google.com/spreadsheets/d/1sGmyr9dZwLhfFWcSgwviwm2X838h0CQF4KWRqX_eXkA/edit#gid=685523897 I have tried unsuccessfully to limit the series up to only 3 Fibonacci numbers. As you see in my shared worksheet, some prime numbers are calculated to 6 or even 7 Fibonacci numbers. I expect that in next research, a simpler formula between these types of numbers can be discovered. All feedback is welcome. Regards,
Thinh Nghiem
Interesting. Do you consider any practical application or your results are just a part of "pure" number theory?
Dear Thin,
Nice observation.
In fact, one can prove the following general result:
Theorem: If n > 1, then n equals a finite sum of Fibonacci numbers.
The proof is simple:
If n is a Fibonacci number, there is nothing to proof
n = Fm = Fm-1 +Fm-2
( by the recurrence of Fibonacci)
If not, choose the closest Fibonacci Fm to n,
that is, n = Fm + n1.
Now if n1 is Fibonacci number , done.
n = Fibonacci + Fibonacci.
If not, proceed the same for n1. choose the closest Fibonacci Fk to n1
n1 = Fk + n2 ,then n = Fm + n1 = Fm + Fk + n2
Keep on the same procedure for n2 in decreasing order to reach the smallest Fibonacci in the sequel and the proof is complete.
Example 1. n = 354 224848179 261915096
The closest Fibonacci number to n is F100 = 354 224848179 261915075
n = F100+ 21
the closest Fibonacci number to 21 is F7 = 13
then n = F100+ 21 = F100+ F7 + 8 = F100+ F7 + F6 .
One May investigate the problem:
What is the smallest sum of Fibonacci numbers that represent a given positive integer n?
Best wishes
@ Przemysław Kowalik
To say honestly, I am doing my research relating to this subject in my free time just for relax. I love mathematics from my childhood. I am not sure if this research can have any practical application. If you think any area can use it (security nerwork, cryptography etc.) please recommend it. thank you
@ Issam Kaddoura
The problem is when we repeat our procedure to all possible Fibonacci numbers, we are not sure the final Fk is Fobonacci or not.
What is the smallest sum of Fibonacci numbers that represent a given positive integer n?
You are right. In part 2, I just analyzed prime number into sum of many Fibonacci numbers. In part 3, I will limit to the smallest sum as much as possible
@ Thinh,
Yes, Fk is a Fibonacci number. It is the closest one to n1 .
(check by example) .The algorithm works.
It is simple to observe [any number has a closest Fibonacci number.]
A good problem is:
Given any positive integer n. What is the minimum number of Fibonacci numbers needed such that n equals their sum?
P.S n needs not prime integer.
Best
If you find the rule for generating rather large prime numbers (i.e. at range near 2^160 or greater) it would aid asymmetric cryptography (for primitives construction based on the discrete logarithm problem and/or elliptic curves).
Can suggest looking through papers related to Fibonacci numbers by Volodymyr Luzhetskyi and/or Alexey Stakhov.
@ Thinh Nghiem
The code of Fibonacci numbers by prime numbers first (as I know) was revealed (shown) by Plichta in 1998 (see: Peter Plichta, Das Primzahlkreuz, Band III, Die 4 Pole der Ewigkeit; page 188 ff.).
Your paper may be new or an expandation; I won’t test it.
But if you are looking for t h e application, Plichta gave it.
There is a relationship between the prime coded fibonacci numbers and the rule of the plan for atomic materials (see at the same place as above named).
In deeply recognition to Peter Plichta, best regards, Peter Kepp
@ Thinh Nghiem
Sorry, as fare as I know there is no link to that `article´ because it is a chapter of a book.
The ISBN of that book (in German!):
ISBN 3—9802808—2—9
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