[seqfan] Re: An arithmetic conjecture

[Tanya Khovanova]

I am definitely interested.


--- On Fri, 3/6/09, David Wilson <davidwwilson at comcast.net> wrote:

> From: David Wilson <davidwwilson at comcast.net>
> Subject: [seqfan] Re: An arithmetic conjecture
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Date: Friday, March 6, 2009, 12:26 AM
> To my knowledge, no one knows how to prove your statement.
> 
> Without getting heavily into it, this problem belongs to a
> class of problems 
> with problems like:
> 
> Does every sufficiently large power of 2 include the digit
> 0 in base 10?
> 
> which statistically are true with probability 1, but have
> not to my 
> knowledge been proved.
> 
> I could go more deeply into this problem. I have a general
> conjecture 
> implying that all sufficiently large m satisfy [2^m / 3^k]
> mod 6 = 3 for 
> some k > 0, but I cannot prove this conjecture nor can I
> vouch for m = 26 as 
> the largest exception.
> 
> Explaining my conjecture is too involved for a late night
> email. I might go 
> into it if there is sufficient interest from seqfan (or
> Tanya).
> 
> ----- Original Message ----- 
> From: "Peter Luschny"
> <peter.luschny at googlemail.com>
> To: <seqfan at list.seqfan.eu>
> Sent: Thursday, March 05, 2009 1:31 PM
> Subject: [seqfan] An arithmetic conjecture
> 
> 
> > Dear all,
> >
> > an arithmetic conjecture from an old discussion
> > in the newsgroup de.sci.mathematik:
> >
> > ============== Conjecture ===================
> >
> > For all m > 26 there exist a k > 0 such that
> >         [2^m / 3^k] mod 6 = 3.
> >
> > =============================================
> >
> > Can someone give a proof?
> >
> > The attempt of numerical falsification gives
> > rise to two sequences, defined by:
> >
> > Start M[1] = 1, K[1] = 0. For given m > 1
> > let s(m) denote the smallest k such that
> > the conjecture holds - assuming existence -
> > or 0 if not. Further let t(m) be the maximum
> > of the s(i) for all i <= m.
> >
> > Now list those pairs m,t(m) where t(m) increases.
> >
> > M : 1, 5, 8, 19, 21, 27, 49, 110, 118, 165, 2769,
> 2837, 3661, 14354,
> > 59913, 2712849,
> > K : 0, 2, 3,  5, 12, 15, 21,  29,  34,  58,   61,  
> 65,   70,    74,
> > 103,     121,
> >
> > To paraphrase the evidence of these sequences:
> > "You can quickly find such a k, even for large
> m."
> >
> > Can someone extend the sequences?
> >
> > Cheers Peter
> >
> > Maple, in the region of conjectured validity:
> > maxK := 1; pow2 := 2^26;
> > for m from 27 to 1000 do
> >  k := 1; pow3 := 3; pow2 := pow2 + pow2;
> >  while modp(iquo(pow2,pow3),6) <> 3 do
> >        pow3 := 3*pow3: k := k+1; od;
> >  if k > maxK then maxK := k; print(m,maxK); fi;
> > od:
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> 
> 
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