[seqfan] Re: An arithmetic conjecture

[David Wilson]

That's always been my problem. The JIS is TeX only now.

----- Original Message ----- 
From: "Martin Fuller" <martin_n_fuller at btinternet.com>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Sent: Tuesday, March 24, 2009 1:46 PM
Subject: [seqfan] Re: An arithmetic conjecture


>
> David Wilson,
>
> I would like to publish, but I do not know how or where.  Would you be 
> interested in co-authoring?  I could do most of the writing and 
> type-setting, and you would provide advice and direction.
>
> Martin Fuller
>
> --- On Fri, 20/3/09, David Wilson <dwilson at gambitcomm.com> wrote:
>
>> From: David Wilson <dwilson at gambitcomm.com>
>> Subject: [seqfan] Re: An arithmetic conjecture
>> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
>> Date: Friday, 20 March, 2009, 9:21 PM
>> No, I think you're right.
>>
>> I was mistaken that the number of nth base-b zeroless
>> number was
>> approximately exponential in n (c^n), when it is actually
>> polynomial in
>> n (n^c). That makes all the difference.
>>
>> I redid the math, and your k value dropped out of it.
>>
>> So I rescind my conjecture about k-automatic sequences in
>> deferences to
>> your work. You really should publish.
>>
>> I'll take a look at palindromes when I can.
>>
>> Martin Fuller wrote:
>> >> From: David Wilson <dwilson at gambitcomm.com>
>> >> Subject: [seqfan] Re: An arithmetic conjecture
>> >> To: "Sequence Fanatics Discussion list"
>> <seqfan at list.seqfan.eu>
>> >> Date: Tuesday, 17 March, 2009, 2:46 PM
>> >>
>> > [cut]
>> >
>> >> I have a conjecture along these lines:
>> >>
>> >> If two bases a >= 2 and b >= 2 and two sets
>> of
>> >> integers A and B where
>> >>
>> >>     a and b are not powers of the same integer
>> (e.g, a = 4,
>> >> b = 8 is
>> >> unacceptable)
>> >>     A and B are infinite,
>> >>     A and B have limit density 0 over the
>> integers,
>> >>     A is a-automatic (the base-a representations
>> of the
>> >> elements of A
>> >> form a regular language) and B is b-automatic.
>> >>
>> >> Then A and B have finite intersection.
>> >>
>> >> Example:
>> >>
>> >> Let a = 2, b = 10, A = powers of 2, B = numbers
>> with no 0
>> >> in their
>> >> base-10 numerals.
>> >>
>> >> This example easily conforms to the first three
>> conditions.
>> >> A is 2-automatic, with its base-2 numerals forming
>> the
>> >> regular language 10*.
>> >> B is 10-automatic, with is base-10 numerals
>> forming the
>> >> regular language
>> >> [123456789]+
>> >>
>> >> My conjecture implies that A and B have finite
>> >> intersection, that is,
>> >> there are a finite number of powers of 2 without
>> zeroes in
>> >> their base-10
>> >> representations.
>> >>
>> >> My conjecture also implies your conjecture.
>> >>
>> >>
>> > [cut]
>> >
>> > Some counter-conjectures (using your reasoning from 7
>> March 2009):
>> >
>> > A,B = numbers without a zero in base a,b
>> > I conjecture that the intersection is infinite for any
>> pair a,b >= 3
>> > Example: a=3, b=4
>> > The sequence starts 1, 2, 5, 7, 13, 14, 22, 23, 25,
>> 26, 41, 43, 53, 121, 122, 125, 149, 151, 157, 158, 214, 215,
>> 229, 230, 233, 238, 239, 365, 367, 373, 374, 377, 445, 446,
>> 473, 475, 485, 607, 617, 619, 634, 635, 637, 638, 697, 698,
>> 701, 725, 727 (not in OEIS)
>> > The number of elements up to n should be O(n^k) with k
>> = log(2)/log(3) + log(3)/log(4) - 1, approximately k = 0.42.
>>  Up to 10^13 the constant is around 3.
>> >
>> > [
>> > Further conjectures:
>> > Numbers without a zero in all bases 3..10: infinite
>> > Numbers without a zero in all bases 3..15:
>> largest=17392214961514563152363
>> > Numbers without a zero in all prime bases
>> 3<=p<=10^10: infinite
>> > Numbers without a zero in all prime bases
>> 3<=p<=10^100: finite
>> > The behaviour is controlled by the sign of:
>> > k = 1-sum{bases b}(1-log(b-1)/log(b))
>> > Any help estimating this function for odd primes?
>> > ]
>> >
>> > A,B = palindromes in base a,b
>> > Are palindromes k-automatic?
>> > I conjecture that any pair of bases that are not
>> powers of the same integer give rise to an infinite
>> sequence.  The distribution is O(log(n)) in each case.
>> > Examples in OEIS: bases 2&10 A007632 (& see
>> links), bases 2&3 A060792, bases 3&4 to 6&7
>> A097928 to A097931, bases 7&8 A099145, bases 8&9
>> A099146.
>> >
>> > Have I missed something?
>> >
>> > Martin Fuller
>> >
>> >
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
>> >
>> >
>> >
>>
>>
>>
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>>
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>
>
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>
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