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A006705 Solution to Pellian: y such that x^2 - n y^2 = +- 1, +- 4.
(Formerly M0398)
5
0, 1, 1, 0, 1, 2, 3, 1, 0, 1, 3, 1, 1, 4, 1, 0, 1, 4, 39, 1, 1, 42, 5, 1, 0, 1, 5, 3, 1, 2, 273, 1, 4, 6, 1, 0, 1, 6, 4, 1, 5, 2, 531, 3, 1, 3588, 7, 1, 0, 1, 7, 5, 1, 66, 12, 2, 20, 13, 69, 1, 5, 8, 1, 0, 1, 8, 5967, 1, 3, 30, 413, 2, 125, 5, 3, 39, 1, 6, 9, 1, 0, 1, 9, 6, 1, 1122, 3, 21, 53 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

When n is a square, the trivial solution (x,y) = (1,0) is taken; otherwise we take the least nontrivial solution that satisfies one of the four equations with +1, -1, +4 or -4. - Ray Chandler, Aug 22 2015

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

Robert G. Wilson v and Ray Chandler, Table of n, a(n) for n = 1..10000 (first 150 terms from Robert G. Wilson v)

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)

MATHEMATICA

r[x_, n_] := Reduce[lhs = x^2 - n*y^2; y > 0 && (lhs == -1 || lhs == 1 || lhs == -4 || lhs == 4), y, Integers]; a[n_ /; IntegerQ[Sqrt[n]]] = 0; xx[n_ /; IntegerQ[Sqrt[n]]] = 1; a[n_] := (x = 1; While[r[x, n] === False, x++]; xx[n] = x; y /. ToRules[r[x, n]]); A006705 = Table[yn = a[n]; Print[{n, xx[n], yn}]; yn, {n, 1, 65}] (* Jean-François Alcover, Mar 08 2012 *)

CROSSREFS

Cf. A006704.

Sequence in context: A321897 A050074 A278313 * A031269 A006703 A133623

Adjacent sequences:  A006702 A006703 A006704 * A006706 A006707 A006708

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

3 terms corrected by Jean-François Alcover, Mar 09 2012

Extended by Ray Chandler, Aug 22 2015

STATUS

approved

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Last modified September 15 14:32 EDT 2019. Contains 327078 sequences. (Running on oeis4.)