A082394 -id:A082394 - cp's OEIS frontend

A082393 Let p = n-th prime of the form 4k+1, take the integer solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with the smallest y >= 1; sequence gives value of y.

4, 180, 8, 1820, 12, 320, 9100, 226153980, 267000, 53000, 6377352, 20, 15140424455100, 113296, 519712, 2113761020, 3726964292220, 190060, 183567298683461940, 448036604040, 28, 386460, 70255304, 649641205044600

Offset: 1

Cino Hilliard, Apr 14 2003

			For n = 1, p = 5, x=9, y=4 since 9^2 = 5*4^2 + 1, so a(1) = 4.

C. Stanley Ogilvy, Tomorrow's Math, 1972, p. 119.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Values of x are in A081232. Cf. A082394, A081233, A081234. Equals A002349(p).

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cf]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; t = {}; Last /@ PellSolve /@ Select[Prime@Range@54, Mod[ #, 4] == 1 &] (* Robert G. Wilson v, Feb 28 2006 *)

p4xp1(n,m) = { forstep(p=1,m,4, for(y=1,n, if(isprime(p), x=y*y*p+1; if(issquare(x), print1(y" "); break; ) ) ) ) }

More terms from Robert G. Wilson v, Feb 28 2006

A081231 Let p = n-th prime of the form 4k+3, take smallest solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and y >= 1; sequence gives value of x.

Original entry on oeis.org

2, 8, 10, 170, 24, 1520, 3482, 48, 530, 48842, 3480, 80, 82, 227528, 962, 4730624, 10610, 77563250, 1728148040, 64080026, 168, 4190210, 8994000, 16266196520, 278354373650, 224, 226, 6195120, 3674890, 139128, 115974983600, 138274082

Offset: 1

N. J. A. Sloane, Apr 18 2003

			For n=3, p = 11, x=10, y=3 since we have 10^2 = 11*3^2 + 1, so a(3) = 10.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Values of y are in A082394. Equals A002350(p). Cf. A082393.

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cf]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; Transpose[ PellSolve /@ Select[ Prime[ Range[72]], Mod[ #, 4] == 3 &]][[1]] (* Robert G. Wilson v, Sep 02 2004 *)

More terms from Robert G. Wilson v, Sep 02 2004

A081232 Let p = n-th prime of the form 4k+1, take smallest solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and y >= 1; sequence gives value of x.

Original entry on oeis.org

9, 649, 33, 9801, 73, 2049, 66249, 1766319049, 2281249, 500001, 62809633, 201, 158070671986249, 1204353, 6083073, 25801741449, 46698728731849, 2499849, 2469645423824185801, 6224323426849, 393, 5848201, 1072400673

Offset: 1

N. J. A. Sloane, Apr 18 2003

			For n = 1, p = 5, x=9, y=4 since 9^2 = 5*4^2 + 1, so a(1) = 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Values of y are in A082393. Cf. A082394, A081233. Equals A002350(p).

PellSolve[(m_Integer)?Positive] := Module[{cof, n, s}, cof = ContinuedFraction[Sqrt[m]]; n = Length[Last[cof]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ContinuedFraction[Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; First /@ PellSolve /@ Select[Prime@Range@54, Mod[ #, 4] == 1 &] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Feb 28 2006

A173202 Solutions y of the Mordell equation y^2 = x^3 - 3a^2 + 1 for a = 0,1,2, ... (solutions x are given by the sequence A000466).

Original entry on oeis.org

0, 5, 58, 207, 500, 985, 1710, 2723, 4072, 5805, 7970, 10615, 13788, 17537, 21910, 26955, 32720, 39253, 46602, 54815, 63940, 74025, 85118, 97267, 110520, 124925, 140530, 157383, 175532, 195025, 215910, 238235, 262048, 287397, 314330, 342895

Offset: 1

Michel Lagneau, Feb 12 2010

For many values of k for the equation y^2 = x^3 + k, all the solutions are known. For example, we have solutions for k=-2: (x,y) = (3,-5) and (3,5). A complete resolution for all integers k is unknown. Theorem: Let k be < -1, free of square factors, with k == 2 or 3 (mod 4). Suppose that the number of classes h(Q(sqrt(k))) is not divisible by 3. Then the equation y^2 = x^3 + k admits integer solutions if and only if k = 1 - 3a^2 or 1 - 3a^2 where a is an integer. In this case, the solutions are x = a^2 - k, y = a(a^2 + 3k) or -a(a^2 + 3k) (the first reference gives the proof of this theorem). With k = -1 - 3a^2, we obtain the solutions x = 4a^2 + 1, y = a(8a^2 + 3) or -a(8a^2 + 3). For the case k = 1 - 3a^2, we obtain the solution x = 4a^2 - 1 given by the sequence A000466.

			With a=3, x = 35 and y = 207, and then 207^2 = 35^2 - 26.

T. Apostol, Introduction to Analytic Number Theory, Springer, 1976
D. Duverney, Theorie des nombres (2e edition), Dunod, 2007, p.151

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
W. J. Ellison, F. Ellison, J. Pesek, C. E. Stall & D. S. Stall, The diophantine equation y^2 + k = x^3, J. Number Theory 4 (1972), 107-117.
Helmut Richter, Solutions of Mordell's equation y^2 = x^3 + k (solutions for 0
School of Mathematics and Statistics, University of St Andrews, Louis Joel Mordell.
Eric Weisstein's World of Mathematics, Mordell Curve.
D. J. Wright, Mordell's Equation.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Diophantine equations: see also Pellian equation: (A081233, A081234), (A081231, A082394), (A081232, A082393); Mordell equation: A053755, A173200; Diophantine equations: A006452, A006451, A006454.

I:=[0, 5, 58, 207]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 02 2012

for a from 0 to 100 do : z := evalf(a*(8*a^2 - 3)) : print (z) :od :

CoefficientList[Series[x*(5+38*x+5*x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 02 2012 *)
CoefficientList[Series[E^x (5 x + 24 x^2 + 8 x^3), {x, 0, 40}], x]*Table[n!, {n, 0, 40}] (* Stefano Spezia, Dec 04 2018 *)

y = a*(8*a^2 - 3).
a(n) = sqrt(A000466(n)^3 - A080663(n)). - Artur Jasinski, Nov 26 2011
From Colin Barker, Apr 26 2012: (Start)
a(n) = 8*n^3 - 24*n^2 + 21*n - 5.
G.f.: x^2*(5 + 38*x + 5*x^2)/(1 - x)^4. (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 02 2012
E.g.f.: exp(x)*(5*x + 24*x^2 + 8*x^3). - Stefano Spezia, Dec 04 2018

A202280 y-values in the solutions to x^2 - 151*y^2 = 1.

Original entry on oeis.org

0, 140634693, 486075138127903440, 1680019594496931198149880507, 5806645138782932871709060110684609120, 20069484831126506853711147766546649419638369093, 69366081749442007616768328339033006654521310003044446320

Offset: 1

Arkadiusz Wesolowski, Dec 15 2011

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1996, p. 248.

Index entries for linear recurrences with constant coefficients, signature (3456296080, -1).

Cf. A041277, A082394.

LinearRecurrence[{3456296080, -1}, {0, 140634693}, 7]

a(n) = 3456296080*a(n-1) - a(n-2) with a(1) = 0 and a(2) = 140634693.

G.f.: 140634693*x^2/(1 - 3456296080*x + x^2).

cp's OEIS Frontend

A082393 Let p = n-th prime of the form 4k+1, take the integer solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with the smallest y >= 1; sequence gives value of y.

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A081231 Let p = n-th prime of the form 4k+3, take smallest solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and y >= 1; sequence gives value of x.

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A081232 Let p = n-th prime of the form 4k+1, take smallest solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and y >= 1; sequence gives value of x.

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A173202 Solutions y of the Mordell equation y^2 = x^3 - 3a^2 + 1 for a = 0,1,2, ... (solutions x are given by the sequence A000466).

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A202280 y-values in the solutions to x^2 - 151*y^2 = 1.

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