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Clarifying my question

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  • Hugo Scolnik (fiber)
    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
    Message 1 of 3 , Nov 3, 2005
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      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)

      Exambles:

      1 + 5*t gives

      x = 1 x2 = 1
      x = 4 x2 = 16
      x = 6 x2 = 36
      x = 9 x2 = 81
      x = 11 x2 = 121
      x = 14 x2 = 196
      x = 16 x2 = 256
      x = 19 x2 = 361
      x = 21 x2 = 441
      x = 24 x2 = 576
      x = 26 x2 = 676
      x = 29 x2 = 841 etc, all of them are given by


      f = 1 + 10*i + 25*i**2


      f = 16 + 40*i + 25*i**2


      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

      Gordon Chalmers led me to his interesting paper concerning factorization but I did not clearly see its connection with the problem of determining the existence and the number of the quadratic polynomials.

      Any clues ?

      Best

      Hugo Scolnik

      Experience is that marvelous thing that enables you to recognize a mistake when you make it again.


      [Non-text portions of this message have been removed]
    • 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 2 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 3 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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          >
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