A Collatz anomaly (3n+a)
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From: Danny73 on 9 Jul 2010 00:31 On Jul 8, 11:46 pm, Danny73 <fasttrac...(a)att.net> wrote: > On Jul 8, 9:51 pm, Mensanator <mensana...(a)aol.com> wrote: > > > > > > > On Jul 8, 11:07 am, Danny73 <fasttrac...(a)att.net> wrote: > > > > On Jul 8, 1:01 am, Mensanator <mensana...(a)aol.com> wrote: > > > > > On Jul 7, 3:01 pm, Danny73 <fasttrac...(a)att.net> wrote: > > > > > > Where a = [41,43,61,107,113,257,739...] > > > > > > The two restrictions on (a) is-- > > > > > (a) can't = (n) and > > > > > (a) can't be a multiple of (n) > > > > > > Then 3n+a will always loop 4,2,1,4,2,1... > > > > > So? > > > > > with > > > > > > the above list of primes for any (n) other than > > > > > the two restrictions on (a). > > > > > What does that have to do with the price of tea in China? > > > > > Are you claiming there are numbers thay DON'T loop on 4,4,1? > > > > > > Can the next prime on this list be found through > > > > > some algorithm without testing? > > > > > > Also, do all these integers have to be prime? > > > > > Do you not ungerstand the probem or are you just having problems > > > > rxpressing yourself? > > > > > > Dan- Hide quoted text - > > > > > - Show quoted text - > > > > I understand the problem! > > > OK, but the next question does not seem to follow from > > such understanding. > > > > I am just asking do these special integers have to be prime? > > > In order For what? To lead to the anomoly of having the > > sequence end in the loop 4,2,1? > > Yes in this special case it is an anomoly because only > these few primes (a) when added to 3n + (a) will loop 4,2,1 FOR ALL > (n) except if a= n or (a) is a multiple of n. I am using (a) variable > here because I am not sure if all in this list will be prime. > At least the first few that I found are all prime but that does > not mean that as this list continues they will all be prime. > This list will probably --->oo but have no proof of that either. > The latest find is 821 and 1307. Both prime. > Would'nt it be something if only primes would work. > Find one that is not prime where the restrictions on (a) relating > too (n) is met and I will close my questioning. > Is that fair enough? > > > First of all, it's not an anomoly that ANY sequence ends tgat way > > since ALL of them do. > > > Furthermore, what do ypu think the significance of the numbers being > > prime > > ia. Or for that matter, multiples of n? None of these criteria > > will have any bearing whatsoever on whether or nt the sequence ends in > > 4,2,1. > > > Perhaps ther is some interestin gyrations that > > occur when said criteria are met, but you didn't ask that. > > Like I said above, meet the restrictions on (a) relation to (n) > and find a non prime in my list. > > > > > > > > > You haven't even explained what the "anonoly" was. > > > Cdrtin questions actuall DO have reasons why certain properties exist, > > for eample, > > why Mersenne numbers typically hve longer sequences than random > > numbers of similar size (even though > > all end in 4,2,1.) > > > > Also how can the next integer be found without trial and error? > > > I just find this problem tied to 3n+1\3n+a interesting. > > > BTW what is the next integer (prime) (a) that follows 739? > > > > Dan- Hide quoted text - > > > - Show quoted text -- Hide quoted text - > > > - Show quoted text - > > Thanks for the input.- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - It was a little ambitious of me to say for all (n) in this problem. (61) fails to meet the all (n) on the third cycle where n=175. At that point for (n) the sequence goes into a non trivial loop. 41,43 where testing 10 cycles for each and are ok, meaning all sequences except the 10 exceptions for each, loop back too 4,2,1 where the final (n)=410 for a=(41) and final (n)=430 for a=(43). Checking the others out now. Seeing this happening to (61) it is possible that none of these primes could hold for all (n). Maybe if these primes just passed a 2 cycle test it would still be interesting. Dan |