[seqfan] Re: Your A051217 Nonnegative numbers of the form 6^x-y^2.
Moshe Levin
moshe.levin at mail.ru
Mon Nov 14 05:22:09 CET 2011
Yes, and this'd better made by some EiC.
And the same about other "similars"?
ML
14 ноября 2011, 02:38 от franktaw at netscape.net:
> Nevertheless, that fact that it has not been proved that all numbers
> not currently in A051217 do not belong there should be documented in
> the sequence.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: David Wilson <davidwwilson at comcast.net>
>
> I am going to copy your question and my reply to the seqfan discussion
> group for comment.
>
> A051217 is well-defined, it consists of all positive integers n (stated
> to be positive, integers since they are
> in an integer sequence) which are of the form 6^x - y^2 (with x and y
> integer, since we can infer from
> context that 6^x - y^2 is a Diophantine expression).
>
> The real question is whether the sequence is correct as given.
> Specifically, could I have missed values?
>
> To be strictly honest, A051217 actually contains all the numbers in the
> published range (0 <= n <= 1152)
> of the form 6^x - y^2 with 0 <= x <= 1000. No, I did not check all x up
> to infinity. The range I computed
> does not include any b-file values, which would have been added later
> by
> someone else.
>
> For even x, we can choose y = 6^(x/2) which gives the sequence value
> 6^x
> - y^2 = 0. The next larger value
> is gotten by choosing y = 6^(x/2) - 1, giving 6^x - y^2 = 2*6^x - 1.
> For x > 3, we have 2*6^x - 1 > 1152,
> so no even x >= 4 can contribute any values to the published A051217.
>
> For odd x, we observe that numbers of the form 6^x - y^2 are about
> 2*6^(x/2) apart near 0. This means
> that for large x, the probability that some 6^x - y^2 will fall in the
> range 0 <= n <= 1152 is around
> 1152 / (2*6^(x/2)). I computed all the values for x <= 1000, for x >=
> 1001, this formula indicates that
> the probability of finding another element in 0 <= n <= 1152 is in the
> ballpark of 10^-387.
>
> So, technically, yes, I could have missed a value in A051217. However,
> if I had to choose between playing
> these two games:
>
> Game 1: You and I each choose a random subatomic particle from the
> observable universe. If we choose
> the same particle, I go to Hell.
>
> Game 2: If A051217 is missing an element on the published range, I go
> to
> Hell.
>
> I would play game 2.
>
> On 11/10/2011 2:30 AM, Moshe Levin wrote:
> >
> > Dear David,
> >
> > Is A051217 (and similars) well-defined?
> >
> > Are we sure that, e.g., some integer in ]812, 855[ is not expressible
> > as 6^x-y^2?
> >
> > Thanks,
> > ML
> >
> > %%%%%%%%%%%%%%%%%%%%%%%%%%
> >
> > A051217 Nonnegative numbers of the form 6^x-y^2.
> >
> > {0, 1, 2, 5, 6, 11, 20, 27, 32, 35, 36, 47, 71, 72, 95, 116, 135,
> 140,
> > 152, 167, 180, 191, 200, 207, 212, 215, 216, 272, 335, 380, 396, 431,
> > 455, 512, 551, 567, 620, 671, 720, 767, 812, 855, 860, 887, 896, 935,
> > 972, 1007, 1040, 1052, 1071, 1100, 1127, 1152}
> >
>
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