# The relationships between Prime number and Fibonacci number (Part 2)

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This is the upgrading of my first research with the same subject, which can be found in

My conclusion in the first article has been optimized as below:

A whole number, odd or even, can be analyzed as the sum of a prime number and two Fibonacci numbers. Here we have some examples

52362 = 52127 (prime) + 2 (Fibonacci 1) + 233 (Fibonacci 2)

52381 = 51769 (prime) + 2 (Fibonacci 1) + 610 (Fibonacci 2)

Also, this time I verified with range [5 - 10,000,000 (ten millions)]. It’s lucky that no failing case as the first time. That means all positive integer number from 5 to 10,000,000 match my rule above.

To view my result, first of all, readers can go to this location

Here the first column is the prime number, the second column is the first Fibonacci number, the third column is the second Fibonacci number. And the last one is the whole number to be calculated.

Welcome all of you

Regards,

Thinh Nghiem from Vietnam

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Posted (edited)

You wrote that "0.97% failed due to the limitation of great number processing of my programming language (C and Java) ".

Advice is: you can use "long long" for 64 bit signed and "unsigned long long" for 64 bit unsigned integers. They will greatly enhance your integer range. You will be able to find primes from 2 to 2^64-1.

Edited by Sensei

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There are also arbitrary (unlimited) precision arithmetic libraries you can use so the size of numbers is only limited by the storage available.

However, it would be better to try and prove the conjecture.

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23 hours ago, Sensei said:

You wrote that "0.97% failed due to the limitation of great number processing of my programming language (C and Java) ".

Advice is: you can use "long long" for 64 bit signed and "unsigned long long" for 64 bit unsigned integers. They will greatly enhance your integer range. You will be able to find primes from 2 to 2^64-1.

My second attempt to consolidate 3 rules into one whole number = prime + fibonacci1 + fibonacci2 has  been verified with 10,000,000 without failure. Please view my txt file in the link

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On 5/16/2018 at 6:14 PM, theodorenghiem said:

My second attempt to consolidate 3 rules into one whole number = prime + fibonacci1 + fibonacci2 has  been verified with 10,000,000 without failure. Please view my txt file in the link

I feel dyslectic. There is a conjecture suggesting that every natural number $$\geq 2$$ can be written as a sum of a prime and two Fibonacci numbers, is that right?