A006703 Solution to Pellian: y such that x^2 - n*y^2 = +-1.

0, 1, 1, 0, 1, 2, 3, 1, 0, 1, 3, 2, 5, 4, 1, 0, 1, 4, 39, 2, 12, 42, 5, 1, 0, 1, 5, 24, 13, 2, 273, 3, 4, 6, 1, 0, 1, 6, 4, 3, 5, 2, 531, 30, 24, 3588, 7, 1, 0, 1, 7, 90, 25, 66, 12, 2, 20, 13, 69, 4, 3805, 8, 1, 0, 1, 8, 5967, 4, 936, 30, 413, 2, 125, 5, 3, 6630, 40, 6, 9

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443.
• C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
• D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=1..10000
• A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443. (Annotated scanned copy)
• M. Zuker, Fundamental solution to Pell's Equation x^2 - d*y^2 = +-1 [Broken link]

Crossrefs

Cf. A006702 (for the x values), A077233.

Programs

Mathematica

r[x_, n_] := Reduce[ y > 0 && (x^2 - n*y^2 == -1 || x^2 - n*y^2 == 1 ), y, Integers]; a[n_ /; IntegerQ[ Sqrt[n]]] = 0; a[n_] := a[n] = (k = 1; While[r[k, n] === False, k++]; y /. ToRules[r[k, n]]); Table[ Print[ a[n] ]; a[n], {n, 1, 79}] (* Jean-François Alcover, Jan 30 2012 *)
nmax = 500;
nconv = 200; (* The number of convergents'nconv' should be increased if the linear recurrence is not found for some terms. *)
x[n_] := x[n] = Module[{lr}, If[IntegerQ[Sqrt[n]], 1, lr = FindLinearRecurrence[ Numerator[ Convergents[Sqrt[n], nconv]]]; SelectFirst[lr, # > 1 &]/2]];
a[n_] := If[n == 2, 1, SelectFirst[{Sqrt[(x[n]^2 - 1)/n], Sqrt[(x[n]^2 + 1)/n]}, IntegerQ]];
Array[a, nmax] // Quiet (* Jean-François Alcover, Mar 08 2021 *)

Extensions

Corrected and extended by T. D. Noe, May 19 2007