The relationships between Prime number and Fibonacci number

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 either

- 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)

I have tried with the first 1,000 positive integer number from 1 to 1,000 MANUALLY and ensured that all of them matched with one of the two above rules.

I shared my analyzing here in the excel file with 1,000 positive integer number from 1 to 1,000 with the link

https://drive.google.com/file/d/0BzAetX6K_uyAUXZHQTd5V3ZIa2c/view?usp=sharing

The majority of them belong to the first case are formatted with normal writing. I set the minority cases (the second one where result equals to prime minus Fibonacci) with red and bold format.

So prime number and Fibonacci number are in actual not completely independent with each other.

It is perfect if anyone can prove this rule in general case, or explain its reason. I do not think that this is only an accidental effect.

You can discuss here or email me at email deleted

Regards,

Thinh Nghiem