- 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] - --- In primenumbers@yahoogroups.com, "Hugo Scolnik \(fiber\)"

<scolnik@f...> wrote:>

b*t generates perfect squares and when those can be written by a

> I am studying under what cinditions a expression of the form a +

number of quadratic polynomials (obviously a must be a quadratic

residue of b)>

....

> However, 281941 + 510510*t leads to perfect squares like

Basically you want to solve the Diophantine equation x^2 - 510510y -

>

> 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

>

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 = 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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