[seqfan] Re: Your A051217 Nonnegative numbers of the form 6^x-y^2.
franktaw at netscape.net
franktaw at netscape.net
Sun Nov 13 23:37:55 CET 2011
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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