[seqfan] Re: Your A051217 Nonnegative numbers of the form 6^x-y^2.

David Wilson davidwwilson at comcast.net
Sun Nov 13 23:29:44 CET 2011


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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