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Clarifying my question

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  • 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 1 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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