A002349 Take solution to Pellian equation x^2 - n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is a square. A002350 gives values of x.

0, 2, 1, 0, 4, 2, 3, 1, 0, 6, 3, 2, 180, 4, 1, 0, 8, 4, 39, 2, 12, 42, 5, 1, 0, 10, 5, 24, 1820, 2, 273, 3, 4, 6, 1, 0, 12, 6, 4, 3, 320, 2, 531, 30, 24, 3588, 7, 1, 0, 14, 7, 90, 9100, 66, 12, 2, 20, 2574, 69, 4, 226153980, 8, 1, 0, 16, 8, 5967, 4, 936, 30, 413, 2, 267000, 430, 3

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Examples

			For n = 1, 2, 3, 4, 5 solutions are (x,y) = (1, 0), (3, 2), (2, 1), (1, 0), (9, 4).
		

References

• Albert H. Beiler, "The Pellian" (chap 22), Recreations in the Theory of Numbers, 2nd ed. NY: Dover, 1966.
• 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, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• E. E. Whitford, The Pell Equation.

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• 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)
• L. Euler, De solutione problematum diophanteorum per numeros integros, par. 17.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
• E. E. Whitford, The Pell equation, New York, 1912.

Crossrefs

Cf. A002350, A006702, A006703, A006704, A006705. See A033316, A033315, A033319 for records.

Programs

Mathematica

a[n_] := If[IntegerQ[Sqrt[n]], 0, For[y=1, !IntegerQ[Sqrt[n*y^2+1]], y++, Null]; y]
(* Second program: *)
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]}]; f[n_] := If[ !IntegerQ[ Sqrt[n]], PellSolve[n][[2]], 0]; Table[ f[n], {n, 0, 75}]

Python

from sympy.ntheory.primetest import is_square
from sympy.solvers.diophantine.diophantine import diop_DN
def A002349(n): return 0 if is_square(n) else next(b for a,b in diop_DN(n,1)) # Chai Wah Wu, Feb 11 2025

Formula

a(prime(i)) = A081234(i). - R. J. Mathar, Feb 25 2025

Extensions

More terms from Enoch Haga, Mar 14 2002

Better description from Robert G. Wilson v, Apr 14 2003