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November 16th, 2016, 05:17 PM   #1
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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/0BzA...ew?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 theodorenghiem@yahoo.co.nz

Regards,

Thinh Nghiem
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November 17th, 2016, 05:11 AM   #2
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Woah!
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November 17th, 2016, 05:52 PM   #3
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Thank you Hawki. Somebody else found that this rule may not be right for greater integer. I am checking
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November 17th, 2016, 06:16 PM   #4
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November 17th, 2016, 07:28 PM   #5
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Quote:
Originally Posted by thinhnghiem View Post
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/0BzA...ew?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 theodorenghiem@yahoo.co.nz

Regards,

Thinh Nghiem
Yes, but on the other hand when I was in High School I found that you can make any number out of football scores: 2 for a touchback, 3 for a field goal and 7 for a touchdown and extra point. Does that mean there is a special relationship between 2, 3, and 7? No, not really. I'll admit that you have an interesting relationship but it is not a new and wonderful discovery.

-Dan

Addendum: It is generally not a good habit, even on this site, to share your e-mail address openly like this. This is not a criticism, just experience.
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November 20th, 2016, 06:51 PM   #6
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Dear Topsquark,

I also sent my finding to professors in university. They confirmed that it is new. However the problem is nobody can be sure if it is matched for greater values or not. A program running is better than doing manually like me.

Also they recommend me that this falls into the very hard to prove or disprove case.

Anybody have idea how to check more? I am very appreciate to welcome your feedback
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November 29th, 2016, 03:17 AM   #7
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Quote:
Originally Posted by thinhnghiem View Post
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/0BzA...ew?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 theodorenghiem@yahoo.co.nz

Regards,

Thinh Nghiem
This is interesting what i was looking for. You clear my mind now. Thanks for sharing.
Thanks from thinhnghiem
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November 30th, 2016, 01:36 PM   #8
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Originally Posted by AshBox View Post
This is interesting what i was looking for. You clear my mind now. Thanks for sharing.
Thanks AshBox,

We are sharing our learning to grow up with each other.

Do you have coding skills? If yes, it is nice that you can help me in checking with greater number, like 10,000 or 100,000

1,000 is too small to give any conclusion. This is the feedback from other members in community
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November 30th, 2016, 02:02 PM   #9
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Quote:
Originally Posted by thinhnghiem View Post

1,000 is too small to give any conclusion. This is the feedback from other members in community
You don't need more examples you need proof. You'll never prove it if all you do is generate lists.

-Dan
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November 30th, 2016, 06:45 PM   #10
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Quote:
Originally Posted by topsquark View Post
You don't need more examples you need proof. You'll never prove it if all you do is generate lists.

-Dan
You need only one counterexample to disprove it, though.
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