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The relationships between Prime number and Fibonacci number

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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




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


The 4th 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




Thinh Nghiem

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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 by DoMiLu
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