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## Re: Clarifying my question

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• ... b*t generates perfect squares and when those can be written by a number of quadratic polynomials (obviously a must be a quadratic residue of b) ...
Message 1 of 3 , Nov 4, 2005
--- In primenumbers@yahoogroups.com, "Hugo Scolnik \(fiber\)"
<scolnik@f...> wrote:
>
> I am studying under what cinditions a expression of the form a +
b*t generates perfect squares and when those can be written by a
number of quadratic polynomials (obviously a must be a quadratic
residue of b)
>
....
> However, 281941 + 510510*t leads to perfect squares like
>
> x = 2321 x2 = 5387041
> x = 39721 x2 = 1577757841
> x = 62381 x2 = 3891389161
> x = 83479 x2 = 6968743441
> x = 94699 x2 = 8967900601
> x = 99781 x2 = 9956247961
> x = 107789 x2 = 11618468521
> x = 143539 x2 = 20603444521
> x = 154759 x2 = 23950348081
> x = 167849 x2 = 28173286801
> x = 185581 x2 = 34440307561
> x = 196801 x2 = 38730633601
>
> but I could not find quadratic polynomials as before
>

Basically you want to solve the Diophantine equation x^2 - 510510y -
281941 = 0. Plugging the numbers in my Quadratic Diophantine Equation
Solver ( http://www.alpertron.com.ar/QUAD.HTM ) you get all solutions:

x = 510510 u + 2321
y = 510510 u^2 + 4642 u + 10

and also:
x = 510510 u + 39721
y = 510510 u^2 + 79442 u + 3090

and also:
x = 510510 u + 62381
y = 510510 u^2 + 124762 u + 7622

and also:
x = 510510 u + 83479
y = 510510 u^2 + 166958 u + 13650

and also:
x = 510510 u + 94699
y = 510510 u^2 + 189398 u + 17566

and also:
x = 510510 u + 99781
y = 510510 u^2 + 199562 u + 19502

and also:
x = 510510 u + 107789
y = 510510 u^2 + 215578 u + 22758

and also:
x = 510510 u + 143539
y = 510510 u^2 + 287078 u + 40358

and also:
x = 510510 u + 154759
y = 510510 u^2 + 309518 u + 46914

and also:
x = 510510 u + 167849
y = 510510 u^2 + 335698 u + 55186

and also:
x = 510510 u + 185581
y = 510510 u^2 + 371162 u + 67462

and also:
x = 510510 u + 196801
y = 510510 u^2 + 393602 u + 75866

and also:
x = 510510 u + 209891
y = 510510 u^2 + 419782 u + 86294

and also:
x = 510510 u + 240559
y = 510510 u^2 + 481118 u + 113354

and also:
x = 510510 u + 245641
y = 510510 u^2 + 491282 u + 118194

and also:
x = 510510 u + 253649
y = 510510 u^2 + 507298 u + 126026

and also:
x = 510510 u + 256861
y = 510510 u^2 + 513722 u + 129238

and also:
x = 510510 u + 264869
y = 510510 u^2 + 529738 u + 137422

and also:
x = 510510 u + 269951
y = 510510 u^2 + 539902 u + 142746

and also:
x = 510510 u + 300619
y = 510510 u^2 + 601238 u + 177022

and also:
x = 510510 u + 313709
y = 510510 u^2 + 627418 u + 192774

and also:
x = 510510 u + 324929
y = 510510 u^2 + 649858 u + 206810

and also:
x = 510510 u + 342661
y = 510510 u^2 + 685322 u + 229998

and also:
x = 510510 u + 355751
y = 510510 u^2 + 711502 u + 247906

and also:
x = 510510 u + 366971
y = 510510 u^2 + 733942 u + 263790

and also:
x = 510510 u + 402721
y = 510510 u^2 + 805442 u + 317690

and also:
x = 510510 u + 410729
y = 510510 u^2 + 821458 u + 330450

and also:
x = 510510 u + 415811
y = 510510 u^2 + 831622 u + 338678

and also:
x = 510510 u + 427031
y = 510510 u^2 + 854062 u + 357202

and also:
x = 510510 u + 448129
y = 510510 u^2 + 896258 u + 393370

and also:
x = 510510 u + 470789
y = 510510 u^2 + 941578 u + 434158

and also:
x = 510510 u + 508189
y = 510510 u^2 + 1016378 u + 505878

For example your first solution is included in the first family of
solutions I presented above.

It appears that you couldn't find the families because you stopped
the search too soon. You have to continue with numbers x greater than
510510 to start seeing the families.

You can see the method I used at:
http://www.alpertron.com.ar/METHODS.HTM#Parabol

because this equation is a parabolic one (B^2 - 4AC = 0).

Best regards,

Dario Alpern
Buenos Aires - Argentina
• The quadratics giving the perfect squares of the form x2 = 281941 + 510510*t are f = 5387041 + 2369787420*i + 260620460100*i**2 f =
Message 2 of 3 , Nov 5, 2005
The quadratics giving the perfect squares of the form x2 = 281941 + 510510*t
are

f = 5387041 + 2369787420*i +
260620460100*i**2

f = 65977573321 + 262260218220*i +
260620460100*i**2

since 510510 = 2*3*5*7*11*13*17 the conjecture raised by a colleague about
the number of quadratics and the prime factors is false.

Dario: you were absolutely right. Thanks.

Hugo Scolnik

No man is justified in doing evil on the ground of expediency.

----- Original Message -----
From: "Dario Alpern" <alpertron@...>
Sent: Friday, November 04, 2005 5:08 PM
Subject: [PrimeNumbers] Re: Clarifying my question

> --- In primenumbers@yahoogroups.com, "Hugo Scolnik \(fiber\)"
> <scolnik@f...> wrote:
> >
> > I am studying under what cinditions a expression of the form a +
> b*t generates perfect squares and when those can be written by a
> number of quadratic polynomials (obviously a must be a quadratic
> residue of b)
> >
> ....
> > However, 281941 + 510510*t leads to perfect squares like
> >
> > x = 2321 x2 = 5387041
> > x = 39721 x2 = 1577757841
> > x = 62381 x2 = 3891389161
> > x = 83479 x2 = 6968743441
> > x = 94699 x2 = 8967900601
> > x = 99781 x2 = 9956247961
> > x = 107789 x2 = 11618468521
> > x = 143539 x2 = 20603444521
> > x = 154759 x2 = 23950348081
> > x = 167849 x2 = 28173286801
> > x = 185581 x2 = 34440307561
> > x = 196801 x2 = 38730633601
> >
> > but I could not find quadratic polynomials as before
> >
>
> Basically you want to solve the Diophantine equation x^2 - 510510y -
> 281941 = 0. Plugging the numbers in my Quadratic Diophantine Equation
> Solver ( http://www.alpertron.com.ar/QUAD.HTM ) you get all solutions:
>
> x = 510510 u + 2321
> y = 510510 u^2 + 4642 u + 10
>
> and also:
> x = 510510 u + 39721
> y = 510510 u^2 + 79442 u + 3090
>
> and also:
> x = 510510 u + 62381
> y = 510510 u^2 + 124762 u + 7622
>
> and also:
> x = 510510 u + 83479
> y = 510510 u^2 + 166958 u + 13650
>
> and also:
> x = 510510 u + 94699
> y = 510510 u^2 + 189398 u + 17566
>
> and also:
> x = 510510 u + 99781
> y = 510510 u^2 + 199562 u + 19502
>
> and also:
> x = 510510 u + 107789
> y = 510510 u^2 + 215578 u + 22758
>
> and also:
> x = 510510 u + 143539
> y = 510510 u^2 + 287078 u + 40358
>
> and also:
> x = 510510 u + 154759
> y = 510510 u^2 + 309518 u + 46914
>
> and also:
> x = 510510 u + 167849
> y = 510510 u^2 + 335698 u + 55186
>
> and also:
> x = 510510 u + 185581
> y = 510510 u^2 + 371162 u + 67462
>
> and also:
> x = 510510 u + 196801
> y = 510510 u^2 + 393602 u + 75866
>
> and also:
> x = 510510 u + 209891
> y = 510510 u^2 + 419782 u + 86294
>
> and also:
> x = 510510 u + 240559
> y = 510510 u^2 + 481118 u + 113354
>
> and also:
> x = 510510 u + 245641
> y = 510510 u^2 + 491282 u + 118194
>
> and also:
> x = 510510 u + 253649
> y = 510510 u^2 + 507298 u + 126026
>
> and also:
> x = 510510 u + 256861
> y = 510510 u^2 + 513722 u + 129238
>
> and also:
> x = 510510 u + 264869
> y = 510510 u^2 + 529738 u + 137422
>
> and also:
> x = 510510 u + 269951
> y = 510510 u^2 + 539902 u + 142746
>
> and also:
> x = 510510 u + 300619
> y = 510510 u^2 + 601238 u + 177022
>
> and also:
> x = 510510 u + 313709
> y = 510510 u^2 + 627418 u + 192774
>
> and also:
> x = 510510 u + 324929
> y = 510510 u^2 + 649858 u + 206810
>
> and also:
> x = 510510 u + 342661
> y = 510510 u^2 + 685322 u + 229998
>
> and also:
> x = 510510 u + 355751
> y = 510510 u^2 + 711502 u + 247906
>
> and also:
> x = 510510 u + 366971
> y = 510510 u^2 + 733942 u + 263790
>
> and also:
> x = 510510 u + 402721
> y = 510510 u^2 + 805442 u + 317690
>
> and also:
> x = 510510 u + 410729
> y = 510510 u^2 + 821458 u + 330450
>
> and also:
> x = 510510 u + 415811
> y = 510510 u^2 + 831622 u + 338678
>
> and also:
> x = 510510 u + 427031
> y = 510510 u^2 + 854062 u + 357202
>
> and also:
> x = 510510 u + 448129
> y = 510510 u^2 + 896258 u + 393370
>
> and also:
> x = 510510 u + 470789
> y = 510510 u^2 + 941578 u + 434158
>
> and also:
> x = 510510 u + 508189
> y = 510510 u^2 + 1016378 u + 505878
>
> For example your first solution is included in the first family of
> solutions I presented above.
>
> It appears that you couldn't find the families because you stopped
> the search too soon. You have to continue with numbers x greater than
> 510510 to start seeing the families.
>
> You can see the method I used at:
> http://www.alpertron.com.ar/METHODS.HTM#Parabol
>
> because this equation is a parabolic one (B^2 - 4AC = 0).
>
> Best regards,
>
> Dario Alpern
> Buenos Aires - Argentina
>
>
>
>
>
>
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> The Prime Pages : http://www.primepages.org/
>
>
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>
>
>
>
>
>
>
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