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

Charles Greathouse charles.greathouse at case.edu
Thu Nov 17 20:08:12 CET 2011

Essentially what we're looking for is a power of 6 such that the power
times sqrt(6) is nearly an integer.  I guess it would be good to look
at the base-6 expansion of sqrt(6)?

Charles Greathouse
Case Western Reserve University

On Thu, Nov 17, 2011 at 12:53 PM, David Wilson <davidwwilson at comcast.net> wrote:
> 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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