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November 30th, 2016, 07:29 PM  #11 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,663 Thanks: 653 Math Focus: Wibbly wobbly timeywimey stuff.  
December 10th, 2016, 01:30 PM  #12 
Newbie Joined: Aug 2016 From: Auckland Posts: 10 Thanks: 1 
Hello all, I've come back!!! Sorry for my being invisible few days ago. I focused in programming, and with the help of programming community, now I have proved my theory with the first 10,000,000 (ten millions) whole number from 1 to 10,000,000. My program was written in Java. I can test more with greater number, but currently ten millions is enough. I put my data in a microsoft access file Data.accdb, share it in Google drive with link https://drive.google.com/drive/u/0/f...Uc2ZnkzV2xaTkE I shared full access for everyone. So please com there, open this file and see table data1. They're there. Look at column four 'Sum or Difference' in this table. For cases which belong to the second rule Integer = Prime  Fibonacci, I put comment 'Difference'. Otherwise it's NULL. Please enjoy. Last edited by skipjack; December 16th, 2016 at 05:47 AM. 
December 11th, 2016, 07:54 PM  #13 
Newbie Joined: Aug 2016 From: Auckland Posts: 10 Thanks: 1 
Hello all, Please ignore my output file. It has many incorrect calculations. My program has a bug, and I have to fix it first. Last edited by skipjack; December 16th, 2016 at 05:45 AM. 
December 16th, 2016, 02:34 AM  #14 
Newbie Joined: Aug 2016 From: Auckland Posts: 10 Thanks: 1 
Hello all, I checked my data. All of the incorrect records belong to Prime numbers which are greater than 2.1 billions. it results from the limit of Java in huge number processing. I use the same algorithm to rewrite the program in C. It solve a number of missing records, but not all of them For example 1651 has been expressed successfully as 18446744073709550683 2584 1651.That means the prime number found is over 18 billions billions (Terrible !!!) Finally, there are 9,335,421 numbers from 1 to 10,000,000 are analyzed successfuly. The number of failure is 664,579 records ~ 6.6%. There may be two cases: 1. Either the calculation is out of C language capability so it had to break out. 2. Or my conjecture fails for these records. That means there is no pair of Prime/Fibonacci to fulfill the rules stated by me. New data has been updated into the same sharing location in Google drive (Data.txt and Data.accdb) Thank you. Last edited by skipjack; December 16th, 2016 at 05:45 AM. 
December 16th, 2016, 05:27 AM  #15 
Senior Member Joined: Sep 2007 From: USA Posts: 349 Thanks: 67 Math Focus: Calculus 
I recommend using Sage for something like this. It is a computer algebra system designed to work with very large numbers, has a large amount of functions including number theory, and you can create your own algorithms on the fly without much trouble. It would require learning some Python to create the algorithm to find a counterexample, but the language is relatively easy to learn. Instead of generating large lists, I recommend only searching for a counterexample. It would run substantially faster. I would program it to search only a specific range at a given time so you can keep track of the progress. Since Fibonacci numbers are easy to generate, I would generate one, find the difference between your current integer and the Fibonacci number, and ask Sage if it is prime. If not, move on to the next Fibonacci number. Given that the integer can be the difference between a prime and Fibonacci number, it will be impossible to be 100% certain you found a counterexample since the prime numbers and Fibonacci numbers are unbounded. Last edited by Compendium; December 16th, 2016 at 06:05 AM. 
December 18th, 2016, 02:31 PM  #16 
Newbie Joined: Aug 2016 From: Auckland Posts: 10 Thanks: 1 
Great, thank you Compedium. I will try to use Sage


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fibonacci, number, prime, relationships 
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