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Let p = n-th prime, 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.
+10
6
3, 2, 9, 8, 10, 649, 33, 170, 24, 9801, 1520, 73, 2049, 3482, 48, 66249, 530, 1766319049, 48842, 3480, 2281249, 80, 82, 500001, 62809633, 201, 227528, 962, 158070671986249, 1204353, 4730624, 10610, 6083073, 77563250, 25801741449
MATHEMATICA
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]}]; Table[ PellSolve[ Prime[n]][[1]], {n, 35}] (* Robert G. Wilson v, Jul 22 2005 *) f[n_] := Block[{p = Prime[n]}, FindInstance[x^2 == p*y^2 + 1 && x > 0 && y > 0, {x, y}, Integers][[1, 1, 2]]]; Array[f, 40] (* Robert G. Wilson v, Nov 16 2012 *)
Let p = n-th prime of the form 4k+3, take the solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and smallest y >= 1; sequence gives value of y.
+10
6
1, 3, 3, 39, 5, 273, 531, 7, 69, 5967, 413, 9, 9, 22419, 93, 419775, 927, 6578829, 140634693, 5019135, 13, 313191, 650783, 1153080099, 19162705353, 15, 15, 400729, 231957, 8579, 7044978537, 8219541, 5052633, 957397, 153109862634573, 34443, 19
REFERENCES
C. Stanley Ogilvy, Tomorrow's Math, 1972, p. 119.
EXAMPLE
For n=3, p = 11, x=10, y=3 since we have 10^2 = 11*3^2 + 1, so a(3) = 3.
MATHEMATICA
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 &]][[2]] (* Robert G. Wilson v, Sep 02 2004 *)
PROG
(PARI) p4xp3(n, m) = { forstep(p=3, m, 4, for(y=1, n, if(isprime(p), x=y*y*p+1; if(issquare(x), print1(y" "); break; ) ) ) ) }
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.
+10
4
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
EXAMPLE
For n=3, p = 11, x=10, y=3 since we have 10^2 = 11*3^2 + 1, so a(3) = 10.
MATHEMATICA
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 *)
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.
+10
4
9, 649, 33, 9801, 73, 2049, 66249, 1766319049, 2281249, 500001, 62809633, 201, 158070671986249, 1204353, 6083073, 25801741449, 46698728731849, 2499849, 2469645423824185801, 6224323426849, 393, 5848201, 1072400673
EXAMPLE
For n = 1, p = 5, x=9, y=4 since 9^2 = 5*4^2 + 1, so a(1) = 9.
MATHEMATICA
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 *)
Let p = n-th prime, take smallest solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and y >= 1; sequence gives value of y.
+10
4
2, 1, 4, 3, 3, 180, 8, 39, 5, 1820, 273, 12, 320, 531, 7, 9100, 69, 226153980, 5967, 413, 267000, 9, 9, 53000, 6377352, 20, 22419, 93, 15140424455100, 113296, 419775, 927, 519712, 6578829, 2113761020, 140634693, 3726964292220, 5019135, 13, 190060
MATHEMATICA
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]}]; Table[ PellSolve[ Prime[n]][[2]], {n, 40}] (* Robert G. Wilson v, Jul 22 2005 *)
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)
+10
1
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
COMMENTS
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.
REFERENCES
T. Apostol, Introduction to Analytic Number Theory, Springer, 1976
D. Duverney, Theorie des nombres (2e edition), Dunod, 2007, p.151
FORMULA
y = a*(8*a^2 - 3).
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)
EXAMPLE
With a=3, x = 35 and y = 207, and then 207^2 = 35^2 - 26.
MAPLE
for a from 0 to 100 do : z := evalf(a*(8*a^2 - 3)) : print (z) :od :
MATHEMATICA
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 *)
PROG
(Magma) 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
y-values in the solutions to x^2 - 313*y^2 = 1.
+10
0
0, 1819380158564160, 117124856755987405647781716823680, 7540058082713667504003446125203741470945194284480, 485400601250164750241979240919394389707542655611270208094258863360
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1996, p. 248.
FORMULA
a(n) = 64376241658269698*a(n-1) - a(n-2) with a(1) = 0 and a(2) = 1819380158564160.
G.f.: 1819380158564160*x^2/(1 - 64376241658269698*x + x^2).
MATHEMATICA
LinearRecurrence[{64376241658269698, -1}, {0, 1819380158564160}, 5]
PROG
(Magma) I:=[0, 1819380158564160]; [n le 2 select I[n] else 64376241658269698*Self(n-1)-Self(n-2): n in [1..10]]; // Vincenzo Librandi, May 16 2015
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