The relationships between Prime number and Fibonacci number

[thinhnghiem123]

Dears,

Recently when learning programming language, I accidentally found out an interesting relationship between prime number and Fibonacci number.

That is, a positive integer number can be analyzed as one of three following rules

-          the sum of a prime number and a Fibonacci number

For example

16 = 11 (prime) + 5 (Fibonnaci)

61 = 59 (prime) + 2 (Fibonacci)

-          or a prime number minus a Fibonacci number

For example

59 = 61 (prime) – 2 (Fibonacci)

83 = 227 (prime) – 144 (Fibonacci)

 

-          or a Fibonacci number minus a prime number

For example

1651=196418 (Fibonacci) – 194767 (Prime)

1759=10946 (Fibonacci) – 9187 (Prime)

 

By using programming, I have tried to proof my finding up to 10,000,000 (10 million). Among them, there are 96,634 records ~ 0.97% failed due to the limitation of great number processing of my programming language (C and Java)

 

I put all of my 10,000,000 output records in a microsoft access file Data.accdb, and share it in google drive with link

 

https://drive.google.com/drive/u/0/folders/0BzAetX6K_uyALUc2ZnkzV2xaTkE

 

I shared full access for everyone. You can come there, open this file and see table data1.

 

The 4^th column in the data file expresses the format of the whole number (Fibonacci-Prime, Prime+Fibonacci, Prime-Fibonacci, or fail)

 

I welcome for feedbacks from you

 

Regards,

 

Thinh Nghiem

[DoMiLu]

There are several interesting relations between primes and Fibonacci numbers, but so far there are very few that are mathematically proven (you know, it may take 358 year long for a  seemingly easy mathematical conjecture in number theory to be rigorously shown-- Fermat's last theorem). You can see some of them here.

Your finding if true is astounding, and more astonishing is it if someone can prove it. However, I won't be surprise if it fails, like the notorious Euler's conjecture. 

Edited December 23, 2016 by DoMiLu