[seqfan] Re: Finding numbers represented by indefinite binary quadratic forms

[Neil Sloane]

PS Peter Moses kindly pointed out that I don't
know how to use Mma! 3xy isn't typed 3xy,
but 3 x y

and then everything works!



On Wed, Jun 4, 2014 at 11:11 AM, Neil Sloane <njasloane at gmail.com> wrote:

> I'm having trouble with the Mathematica "Reduce" command.
> It is easy to show by hand that x^2+3xy-3y^2=3 has no solutions in
> integers (**)
> But Reduce[x^2 + 3 xy - 3 y^2 - 3 == 0, {x, y}, Integers]
> does not return "False" (***).
>
> How can one use Mma to decide if x^2+3xy-3y^2=3
> has a solution in integers?
>
> Neil
>
> (**) Proof. x must be divisible by 3, say x=3z,
> so we get 3z^2+3yz-y^2=1; but y^2=-1 mod 3 has no solution.
> (***) Mma returns
>
> ((C[1] | C[2]) ∈ Integers && xy == 1/3 (3 - 9 C[1]^2 + 27 C[2]^2) &&
>    x == 3 C[1] && y == 3 C[2]) || ((C[1] | C[2]) ∈ Integers &&
>    xy == 1/3 (3 - 9 C[1]^2 + 3 (1 + 3 C[2])^2) && x == 3 C[1] &&
>    y == 1 + 3 C[2]) || ((C[1] | C[2]) ∈ Integers &&
>    xy == 1/3 (3 - 9 C[1]^2 + 3 (2 + 3 C[2])^2) && x == 3 C[1] &&
>    y == 2 + 3 C[2]),
>
> which presumably tells us nothing.
>
>
> On Wed, Jun 4, 2014 at 10:52 AM, Giovanni Resta <g.resta at iit.cnr.it>
> wrote:
>
>> On 6/4/2014 4:14 PM, Neil Sloane wrote:
>>
>>> Dave, Thanks, that Mathematica command does indeed seem to do the job! It
>>> gives (as you say) a long list of solutions if solutions exist, and
>>> "false"
>>> if they don't.
>>>
>>
>> Yes, in general the Reduce command applied to quadratic Diophantine
>> equations gives False if no solutions are found, a list of solutions if
>> solutions are in finite number, or one or more solutions in exponential
>> form, something like
>> x == 1/24 (-4 + 2 ((5 - 2 Sqrt[6])^(2 C[1]) + (5 + 2 Sqrt[6])^(2 C[1])))
>> with C[1] integer, if there exist infinite solutions.
>>
>> In case just a few equations have to be solved, there is an online applet
>> by  Dario Alpern at this address:
>>
>> http://www.alpertron.com.ar/QUAD.HTM
>>
>> A nice thing about that page is that it provides also a step-by-step
>> solution and that infinite solutions are represented by recurrences
>> instead of exponential expressions.
>>
>> Giovanni
>>
>>
>>
>>
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>>
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>>
>
>
>
> --
> Dear Friends, I have now retired from AT&T. New coordinates:
>
> Neil J. A. Sloane, President, OEIS Foundation
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>


-- 
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com