A006705 - OEIS

%I M0398 #32 Oct 29 2023 21:17:50

%S 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,

%T 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,

%U 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

%N Solution to Pellian: y such that x^2 - n y^2 = +- 1, +- 4.

%C 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

%D 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.

%D C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.

%D D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Ray Chandler, <a href="/A006705/b006705.txt">Table of n, a(n) for n = 1..10000</a> (first 150 terms from Robert G. Wilson v)

%H A. Cayley, <a href="/A002349/a002349.pdf">Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ...</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443. (Annotated scanned copy)

%t 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 *)

%Y Cf. A006704.

%K nonn,easy,nice

%O 1,6

%A _N. J. A. Sloane_

%E 3 terms corrected by _Jean-François Alcover_, Mar 09 2012

%E Extended by _Ray Chandler_, Aug 22 2015