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

David Wilson davidwwilson at comcast.net
Thu Nov 17 18:53:52 CET 2011

```In the case of this particular sequence, there is no longer any doubt
that the published values are correct.
I wrote a program that, for each 0 <= n <= 1152 (the published range),
solved n = 6^x - y^2 for n in A051217
and found a modulus that prohibited n = 6^x - y^2 for n not in A051217.

Given the success of my program, I am willing to conjecture that this
approach works for any similar sequence:
For each n, we will either find a solution for n, or else a modulus
which prohibits a solution.

It turns out that the smallest solutions of n = 6^x - y^2 for 0 <= n <=
1152 have x <= 5. So x <= n is actually
rather generous.

If you want a better feel for this problem, compute the smallest
possible n = 6^x - y^2 > 0 for each 0 <= x <= 30.
Would you be willing to bet that some x >= 30 will produce n in (0, 100000]?

This problem has a smell of the Goldbach Conjecture. Even though we
cannot prove that all even numbers
n >= 4 have even one Goldbach partition, however, the pattern of the
number of Goldbach partitions of 2n
(see the graph of A001031, Goldbach's comet) strongly indicates that
this is the case. It would be nothing
short of miraculous if some stray point were to fall from the comet to
the ground.

On 11/16/2011 5:11 PM, Vladimir Shevelev wrote:
> I agree with David: the sequence is, of course,  well-defined. But it is clear that without an upper estimate for x it cannot be computable. Since till now such an estimate is unknown, then we should use an empirical conjecture. For example, one can conjecture that, for the n-th term of the sequence, x  not exceeds n. In such case one can  write in the title "a(1)=0; for n>=2, a(n) is the least number>a(n-1) of the form 6^x-y^2 with x<=n  and a(n)=0, if such number does not exist". Then, e.g.,  23  pretending to be a(8)  is verified under condition  6^x-y^2-23=0, x=1,...,8. Now this sequence is  well-defined and well computable. In a comment one can pose a conjecture:" It is a monotone sequence containing all numbers of the form 6^x-y^2".
>
> Regards,
> Vladimir

```

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