mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Miscellaneous Math

Reply
 
Thread Tools
Old 2015-08-21, 03:18   #1
jux
 
jux's Avatar
 
Aug 2015

2×33 Posts
Default Fibonacci number as sum of cubes

Hello everyone!
Hopefully this is the right place to post. A while ago I asked on math.stackexchange if there was a Fibonacci number that was the sum of 2 positive cubes, besides 2. I am hoping you guys might have some new ideas. I am a beginner when it comes to number theory and math, but I am surprised to not be able to find any other research on this.
jux is offline   Reply With Quote
Old 2015-08-21, 12:04   #2
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

26×131 Posts
Default

Quote:
Originally Posted by jux View Post
Hello everyone!
Hopefully this is the right place to post. A while ago I asked on math.stackexchange if there was a Fibonacci number that was the sum of 2 positive cubes, besides 2. I am hoping you guys might have some new ideas. I am a beginner when it comes to number theory and math, but I am surprised to not be able to find any other research on this.
cubes are 0 1 and 8 mod 9 sum any two and you can get 0 1 2 7 or 8 mod 9 every number has a pisano period , however lets look at what places these values mod 9 pop up. the Fibonacci sequence mod 9 repeats:

0,1,1,2,3,5,8,4,3,7,1,8,0,8,8,7,6,4,1,5,6,2,8,1,... so only positions that are 1,2,3,4,7,10,12,14,15,16,19,22,23,0 mod 24 can the sum of two cubes to begin with. and that's what I can come up with right now.

Last fiddled with by science_man_88 on 2015-08-21 at 12:09
science_man_88 is offline   Reply With Quote
Old 2015-08-21, 12:22   #3
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

22·5·373 Posts
Default

Quote:
Originally Posted by jux View Post
Hello everyone!
Hopefully this is the right place to post. A while ago I asked on math.stackexchange if there was a Fibonacci number that was the sum of 2 positive cubes, besides 2. I am hoping you guys might have some new ideas. I am a beginner when it comes to number theory and math, but I am surprised to not be able to find any other research on this.
There are at most finitely many.

Read H. Cohen's book on Diophantine Equations.

Last fiddled with by R.D. Silverman on 2015-08-21 at 12:30
R.D. Silverman is offline   Reply With Quote
Old 2015-08-21, 12:31   #4
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

746010 Posts
Default

Quote:
Originally Posted by R.D. Silverman View Post
There are at most finitely many.

Read H. Cohen's book on Diophantine Equations.
It is possible that an application of Baker's linear forms in logarithms might be able
to place an upper bound on the solutions. I am not an expert in this area.
R.D. Silverman is offline   Reply With Quote
Old 2015-08-21, 13:08   #5
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

100000110000002 Posts
Default

Quote:
Originally Posted by R.D. Silverman View Post
There are at most finitely many.

Read H. Cohen's book on Diophantine Equations.
not to mention you can use that fact that prime positions matter most because starting at 1,1 instead of 0,1 any position that isn't prime divides by a lower prime one and so the value the latter one divided by the previous one needs specific form for the sum of cubes to work ( a cube number after division is the easiest example).
science_man_88 is offline   Reply With Quote
Old 2015-08-21, 14:29   #6
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

164448 Posts
Default

Quote:
Originally Posted by science_man_88 View Post
not to mention you can use that fact that prime positions matter most because starting at 1,1 instead of 0,1 any position that isn't prime divides by a lower prime one and so the value the latter one divided by the previous one needs specific form for the sum of cubes to work ( a cube number after division is the easiest example).
Complete word salad. Total nonsense.
R.D. Silverman is offline   Reply With Quote
Old 2015-08-21, 14:40   #7
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

203008 Posts
Default

Quote:
Originally Posted by R.D. Silverman View Post
Complete word salad. Total nonsense.
not true for example 2 is at a prime position starting with 1,1 therefore with the alternate form of the Fibonacci sequence starting 1,1,2,3,5,8 ( note no 0) if any formula connecting the positions that are multiples of 3 together can be a cube infinitely often then there could be infinitely many of them contrary to what you said earlier. so in order to disprove infinitely many all you have to do is check the prime position ones assuming of course you can find a formula to link the fibonacci numbers that are in composite positions to the prime factors of that position then you have to disprove it can be a cube infinitely often. the only time you have to check positions that aren't prime would be if no formula that can be a cube infinitely often is found.
science_man_88 is offline   Reply With Quote
Old 2015-08-21, 15:12   #8
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

1D2416 Posts
Default

Quote:
Originally Posted by science_man_88 View Post
not true for example 2 is at a prime position starting with 1,1 therefore with the alternate form of the Fibonacci sequence starting 1,1,2,3,5,8 ( note no 0) if any formula connecting the positions that are multiples of 3 together can be a cube infinitely often then there could be infinitely many of them contrary to what you said earlier. so in order to disprove infinitely many all you have to do is check the prime position ones assuming of course you can find a formula to link the fibonacci numbers that are in composite positions to the prime factors of that position then you have to disprove it can be a cube infinitely often. the only time you have to check positions that aren't prime would be if no formula that can be a cube infinitely often is found.
More gibberish.
R.D. Silverman is offline   Reply With Quote
Old 2015-08-21, 15:56   #9
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts
Default

Quote:
Originally Posted by R.D. Silverman View Post
More gibberish.
Quote:
Originally Posted by https://en.wikipedia.org/wiki/Fibonacci_number#Divisibility_properties
Every 3rd number of the sequence is even and more generally, every kth number of the sequence is a multiple of Fk
therefore if there's a form for the composite positioned ones from the primitive ones then you only need to check anything that isn't prime only if there's not a possibility of the prime positioned ones leading to cubes infinitely often.
science_man_88 is offline   Reply With Quote
Old 2015-08-21, 17:14   #10
Batalov
 
Batalov's Avatar
 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

9,391 Posts
Default

science_man_88, stop your

There is nothing that needs to be added to stackexchange's discussion which I am afraid that you were not bothered to read.
Batalov is offline   Reply With Quote
Old 2015-08-21, 17:31   #11
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts
Default

Quote:
Originally Posted by Batalov View Post
science_man_88, stop your

There is nothing that needs to be added to stackexchange's discussion which I am afraid that you were not bothered to read.
I did read it
science_man_88 is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
What is the biggest Fibonacci prime number? George M Lounge 20 2018-01-03 16:04
Squares and Cubes: mfgoode Puzzles 24 2007-08-06 16:20
Prime cubes! fivemack Puzzles 4 2007-07-04 00:18
Fibonacci modulo Fibonacci robert44444uk Math 3 2007-05-19 07:15
Counting Cubes Numbers Puzzles 6 2005-09-03 00:26

All times are UTC. The time now is 13:00.

Tue Apr 20 13:00:24 UTC 2021 up 12 days, 7:41, 0 users, load averages: 3.67, 2.68, 2.30

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.