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Re: Clarifying my question

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  • Dario Alpern
    ... 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
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      --- 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
    • Hugo Scolnik (fiber)
      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
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        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@...>
        To: <primenumbers@yahoogroups.com>
        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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