[seqfan] Re: A039669 and new related numbers

[Tomasz Ordowski]

Robert, there is a mistake at the end of your proof.
So I will write the whole thing in a corrected version.

There are no more of these.  If k - 2^2 = a^2 and k - 2^4 = b^2,
then a^2 - b^2 = 12, and the only nonnegative integer solution of that is
a = 4, b = 2 corresponding to k = a^2 + 2^2 = b^2 + 2^4 = 20, QED.

Best,

Tom Ordo

śr., 21 lut 2024 o 17:38 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

>
> Robert, thanks to you, we have the brilliantly simple
> proof that there are no more such numbers from PS.
>
> Now that this is proven - as I wrote - I can forget about OEIS,
> because the data section {5, 8, 13, 20} is too weak, and...
> the sequence would be of little interest, oh well.
>
> However, my first numbers seem worthy
> of a new sequence on the OEIS. Yes?
> I hear no objection!
>
> Best,
>
> Tom Ordo
>
> śr., 21 lut 2024 o 16:02 <israel at math.ubc.ca> napisał(a):
>
>> There are no more of these.  If k - 2^2 = a^2 and k - 2^4 = b^2,
>> then a^2 - b^2 = 12, and the only nonnegative integer solution of that is
>> a = 4, b = 2 corresponding to k = 16.
>>
>> Cheers,
>> Robert
>>
>> On Feb 21 2024, Tomasz Ordowski wrote:
>>
>>
>> >PS. Similarly. Interesting, but with a poor data section, namely:
>> >Numbers k such that all positive values of k - 2^(2^n) are square,
>> >with natural n > 0. Data: {5, 8, 13, 20}. Please "more", if any...
>> >Note that 2^(2^n) for n > 0 is the square (2^(2^(n-1))^2.
>> >If there are no more terms, what do we do with it?
>> >Prove it and forget about the OEIS. Yes?
>> >
>> >--
>> >Seqfan Mailing list - http://list.seqfan.eu/
>> >
>> >
>>
>>
>> --
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>>
>