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dan73
Posts:
468
From:
ct
Registered:
6/14/08


Re: The tower of Hanoi and the sum of its pieces is related to the primes.
Posted:
Nov 18, 2012 8:20 AM


> The tower of Hanoi and the sum of its pieces > is related to the primes and a sequence in OEIS. > > Rehashing the tower of Hanoi  > > Three pegs where a stack of (n) number of disks > of incremental size have to be moved completely > to another peg one at a time to another peg > without putting a larger disk on a smaller disk. > > The correct number of moves for (n) number > of disks is  > (n) disks correct # of moves > 2 2^21 > 3 2^31 > 4 2^41 > 5 2^51 > 6 etc. > > Now to emphasize the value of each disk the > diameter of the smallest disk and each larger disk. > disk #ndia.area for each disk > > disk #1 = 1.128379167...  1 > disk #2 = 1.5957691216... 2 > disk #3 = 1.9544100476... 3 > disk #4 = 2.2567583341... 4 > etc.. > > Too find the total sum for each (n) stack of > disks is t(n) a triangle number. > Now to sum each piece as it is moved in the > process of 2^n1 moves there are many pieces with the > > same area that are added many times depending > on the size of (n). > > To find this sum  > disks # sum > in tower > 1 disk = 2^23 =1 > 2 disk = 2^34 =4 > 3 disk = 2^45 =11 > 4 disk = 2^56 =26 > 5 disk = 2^67 =57 > 6 disk = 2^78 =120 > 7 disk = 2^89 =247 > etc. > 1,4,11,26,57,120,247... > sequence A000295 in OEIS. > > Now to find out if the number of odd disks are prime > then say checking a (3) disk stack 11/4 = 2.75 = > cf =[2:1,3] > Where the cf will always have just 3 terms and the > last term will equal the # of odd disks to be prime. > In the above case (3). So (3) is prime. > > Checking the 7 disk stack as prime  > 247/120 = 2.0583333333... = cf =[2:17,7]  > so (7) is prime. > > Checked all odd disks stacks from here on except > 0(mod 3 or 5) stacks. It identifies all primes and > composites.
I left out an important fact below that 203 disk stack is term 203 if you refer back to the original generating series above. Also 223 stack is term # 223.
> Checking if a 203 disk stack is prime you have to > first > find the divisor. > 2^203  204 = > 1.2855504354071922204335696738729300820177623950262342 > 682410804e+61 > Then  > 2^204  205 = > 2.5711008708143844408671393477458601640355247900524685 > 364821811e+61 > > 2.5711008708143844408671393477458601640355247900524685 > 364821811e+61 > / > 1.2855504354071922204335696738729300820177623950262342 > 682410804e+61 > = > 2.0000000000000000000000000000000000000000000000000000 > 0000001579090... > cf = > [2:633276076555267103661856982203413833506286894101593 > 23558673,1,10,3,1,1,2] > 203 is not prime because there are more then 3 terms > in the cf. > > Is a 223 disk stack prime? > > 2^223 224 > =1.347997333357531989733350754350981533681857221127028 > 6240551805124384e+67 > 2^224 225 > =2.695994666715063979466701508701963067363714442254057 > 2481103610248991e+67 > 2.0000000000000000000000000000000000000000000000000000 > 000000000000165430594320658993803... > cf = > [2,604483109128938111988049665628242840216079471357411 > 93903819753921,223] > 223 is prime > > I always had a hunch that the tower of Hanoi might > be > related to the primes! > > Dan



