A033315 Incrementally largest values of minimal x satisfying Pell equation x^2 - D*y^2 = 1.

1, 3, 9, 19, 649, 9801, 24335, 66249, 1766319049, 158070671986249, 2469645423824185801, 159150073798980475849, 838721786045180184649, 25052977273092427986049, 3879474045914926879468217167061449

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Links

• Ray Chandler, Table of n, a(n) for n = 1..62 (first 50 terms from T. D. Noe)
• Eric Weisstein's World of Mathematics, Pell Equation.

Crossrefs

Records in A033313 (or A002350).

Cf. A033313, A033314, A002349, A002350.

Cf. A033316 (corresponding values of D).

Programs

Mathematica

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[Sqrt[m]]; n = Length[Last[cf]]; If[n == 0, Return[{}]]; If[OddQ[n], n = 2 n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
xx = DeleteCases[PellSolve /@ Range[10^5], {}][[All, 1]];
Reap[Module[{x, record = 0}, Sow[1]; For[i = 1, i <= Length@xx, i++, x = xx[[i]]; If[x > record, record = x; Sow[x]]]]][[2, 1]] (* Jean-François Alcover, Nov 21 2020, after N. J. A. Sloane in A002349 *)