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[Artur]

Dear Max and Rest of Seqfans

Do you able to proove  my conjecture that form  1+2xy+x^2 y+y^2  don't 
have integer solutions or find one of these solution as countersample?

BEST WISHES
ARTUR


Max Alekseyev pisze:
> On Nov 16, 2007 11:06 AM, Artur <grafix at csl.pl> wrote:
>
>   
>> I'm apologize but but I was do mistake in formula
>> Find number m such that 5n^4(4+n^2) is square
>> Mathemtica sample Code:
>> a={};Do[If[IntegerQ[Sqrt[5n^4 (4+n^2)]],AppendTo[a,n]],{n,1,1364}];a
>>
>> Out: {1,4,11,29,76,199,521,1364}
>>     
>
> If I got you correctly, you're looking for positive integer solutions to
> 5n^4(4+n^2) = m^2
> It implies that n^2 divides m. Let m=n^2*k. Then the equation is equivalent to
> 5(4+n^2) = k^2
> or
> k^2 - 5n^2 = 20.
> This is a generalized Pell equation and you can plug its coefficients
> to Dario Alpern's on-line tool:
> http://www.alpertron.com.ar/QUAD.HTM
> to get solution in the form of recurrent sequences.
>
> With respect to n, the solution consists of the following 3 sequences,
> satisfying the same recurrent relation n(j+1)=18*n(j)-n(j-1) with
> different initial conditions:
>
> n(0) = 1, n(1) = 29: { 1, 29, 521, 9349, 167761, 3010349, ... }
>
> n(0) = 4, n(1) = 76: { 4, 76, 1364, 24476, 439204, 7881196, ... }
>
> n(0) = 11, n(1) = 199:  { 11, 199, 3571, 64079, 1149851, 20633239, ... }
>
> Your sequence is the union of these three.
>
> Max
>
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