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May 26th, 2010, 07:04 PM  #1 
Newbie Joined: May 2010 Posts: 6 Thanks: 0  Need help creating formula for:Fn1 + Fn2 + ... + Fnk
I am a novice computer programmer, and I am looking to write a simple data compression program. It seems that one could use a recurrence relation sequence or Closedform expression thereof  to compress any file down to a few bytes. See files on a computer could be looked at as one really big(astronomical number). A small formula, only a few bytes wide, could ideally represent any abitrary sized file(e.g. 1, 2, 50 gigs+, etc), and when one needed to decompress  the zip program would merely caclulate the formula, and the end result would be a really big binary number. The binary would be your file. Fibonacci numbers do not include every number  which, of course, is a very good thing. So I was thinking that I could use a formula similar to "Binet's formula". . So I need a similar Closedform expression that would calculate: Fn1 + Fn2 + ... + Fnk k being any arbitrary number > 2. [color=#FF0000]My requests[/color]: [color=#FF0000]=================================================[/color] (1) Would anyone be so kind as to write me a closedform expression for the recurrence relation sequence: Fn1 + Fn2 + ... + Fnk  (2) If no one is willing to write me a formula, then could someone at least talk me through writing it myself. Perhaps you could talk me through the motives and puposes behind the different parts of Binet's formula  that would help alot(e.g. Why use the sqrt of 5 in "phi" as opposed to another sqrt anyways). I am not well versed in mathematics, but if you explain things in graphical and simple terms I will eventually figure out what your writing. [color=#FF0000]=================================================[/color] i am all to willing to do this myself except I do not yet have the necessary math expertise to do so. I also, unfortunately, have had a very dibilitating mental disorder for a number of years now and it has put my entire life on ice, I am currently taking medication, and when I get better I will VERY gladly fufill all my mathematical endeavors. If creating such a formula is to much an incredible task then let me know. It seems possibly reasonably doable because the formula would be veyr muc based on Binet's already composed fromula. Thank you to whomever helps. Math is awesome, and good day!!! 
May 27th, 2010, 05:45 AM  #2  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Need help creating formula for:Fn1 + Fn2 + ... + Fnk Quote:
By the Pigeonhole Principle, at most Code: 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000237% Quote:
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May 27th, 2010, 10:32 PM  #3 
Newbie Joined: May 2010 Posts: 6 Thanks: 0  Re: Need help creating formula for:Fn1 + Fn2 + ... + Fnk
Thank you CRGreathouse  your answer was fantastic. My mind is really really foggy right now, as usual, but as soon as I am thinking clearly enough to ask and discuss super compression I will be revisiting this thread. I understand that super compression is considered a pipe dream, but I also understand that anytime anyone ever wanted anything in this world and they sought out all the necessary information and resources to achieve  they eventually miraculously do achieve despite everyone saying their dream or idea was impossible. This phenomena has occurred countless times. What I am saying is that there is always a way............................................... ............... It dawned on me ,as I was reading your post, a number of really good reasons why my formula or "most" any formula or algorithm would not work  some of which I already new. The truth is that I didn't genuinely think that my formula could cover every possible number. I did however overlook the obvious that the range between one number and the next greater gets bigger and bigger towards infinity. I intended to use this formula as a learning experience . For me,there are just to many ins and outs to numbers just to rule out super compression, and even though I see compelling reasons to discount it, I am compelled even more by the infinite possabilities to make a way where there was no way. Very good day to you, and thanks again! 
May 28th, 2010, 05:16 AM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Need help creating formula for:Fn1 + Fn2 + ... + Fnk Quote:
What you need to do is understand the limitations and find ways of working around them. For example, .jpg images achieve extremely high compression by throwing away information that isn't easily visible to the human eye. The Joint Photographic Experts Group understood this result and found a way around it: they don't keep all the original information, just the 'visually significant' information. If you can find a similar way around, you may be able to create a great compression algorithm. But if you simply try to fight the bound, you will fail. (No skin off my back if you try, though.) Consider it this way. There are 2^100 = 1267650600228229401496703205376 100bit files. There are 2^200 = 16069380442589902755419620923411626025222029937827 92835301376 200bit files. If you compress all 200bit files to 100 bits, then there is (by the Pigeonhole Principle  but it's obvious if you think about it) some 100bit file that is the compressed version of at least 2^200 / 2^100 = 1267650600228229401496703205376 different 200bit files. How will you tell which one is the right one to decompress to?  
May 28th, 2010, 03:17 PM  #5  
Newbie Joined: May 2010 Posts: 6 Thanks: 0  Re: Need help creating formula for:Fn1 + Fn2 + ... + Fnk Quote:
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I also think I understand your comment: Quote:
 My problem is that when I look at things like irrational numbers, which may be infinitely complex, I perceive that there is some identity that has not been found yet, and may hold the key to super compression with today's current computer technology. Of course, the more I think about it, I see the seeming impossibility and futility of it. Really, if nothing else, I will consider my endeavors as a really fun learning experience. Very good day to you then, CRGreathouse. Thank you for your help, friend, I've learned alot from your post. [color=#FF0000]EDIT: ================================================== ========[/color] I said earlier that all things were possible, but after thinking about it  what I should have said was that all things are probably possible  if  one has a healthy mind an ability to express that healthy mind, and an infinite amount of resources to work with. However, I now realise the binary representation of numbers on todays current computers is finite. Just thought I'd throw that in.  
May 29th, 2010, 04:29 AM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Need help creating formula for:Fn1 + Fn2 + ... + Fnk Quote:
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But please, continue your investigation  I think you will find it enlightening. I'm glad that you're keeping on open mind.  
May 29th, 2010, 08:44 PM  #7  
Newbie Joined: May 2010 Posts: 6 Thanks: 0  Re: Need help creating formula for:Fn1 + Fn2 + ... + Fnk
I apologize for making another post for my following comments, and I really would have preferred to just edit my previous post up above, but my edit time has expired. So, I don't mean to bump this thread and flag anyone's attention. Quote:
In this following comment, I attempt to correct my earlier comment: Quote:
I apologize again for bumping this thread, I did not want to leave such a mistake unfixed. Also, since I'm posting anyways, I would like to reiterate that, when I'm feeling better, I would still like to present a few theoretical ideas, and ask some hypothetical, and bizzare questions (which may or may not be stupid) of things that have caught my eye about super compression with todays current computer technology. So, I will probably be back. Good day to everyone.  
May 30th, 2010, 01:08 PM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Need help creating formula for:Fn1 + Fn2 + ... + Fnk Quote:
Also (this is deep!), there is a finite limit on the amount of information that can be represented in the entire universe, thanks to the holographic principle. I don't recall the exact amount but it's below a googol bits. Quote:
 
June 1st, 2010, 10:09 PM  #9  
Newbie Joined: May 2010 Posts: 6 Thanks: 0  Re: Need help creating formula for:Fn1 + Fn2 + ... + Fnk
First, to CRGreathouse, I am sorry for any of my ramblings, and lack of haste to respond. I have 0 background in mathematics or really anything at all for that matter (except for a little in C++ programming), and as I've said, I am not thinking coherently. So, I openly admit to and acknowledge my foolishness and ignorance, but I am trying. My response was slow because I was trying to think of a proper and good way to respond, but I don't think my illness will allow this. Quote:
And yes, of course, the limitation is in the representation,  and I meant that our current computer technology is limited  to  our binary representation. Quote:
Well anyway, I am merely rambling, feel free to ignore. Quote:
I guess I figured that since any arbitrary algorithm cannot be, according to your statement:“In the Kolmogorov sense, an irrational number like sqrt(2) has very little information",completely complex that certain or maybe all finite algorithms did not qualify as Kolmogorov complex. I did denote, however, that I was not sure. Now, by "infinitely complex" I meant that certain algorithms can produce an infinite number in which, at least, some newly asymptotically calculated arbitrary parts would break any previous pattern, and so there would be an infinite amount of newness. I was given this impression because of statements like what one might find on Wikipedia, such as this statement listed under "Pi": Quote:
Good day to you CRGreathouse, I have considered it a pleasure to read your writings  
June 2nd, 2010, 07:38 AM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Need help creating formula for:Fn1 + Fn2 + ... + Fnk Quote:
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If that's too much, consider that data requires (a positive amount of) massenergy to store; since the universe's massenergy is finite, its information capacity is finite also. If there was infinite massenergy, then all matter would be attracted with infinite force and hence infinite rapidity (the relativistic version of speed/velocity). This shows that the amount of information in the universe is finite, even if very large. Quote:
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