A077233 -id:A077233 - cp's OEIS frontend

A077232 a(n) is smallest natural number satisfying Pell equation a^2 - d(n)*b^2= +1 or = -1, with d(n)=A000037(n) (a nonsquare). Corresponding smallest b(n)=A077233(n).

1, 2, 2, 5, 8, 3, 3, 10, 7, 18, 15, 4, 4, 17, 170, 9, 55, 197, 24, 5, 5, 26, 127, 70, 11, 1520, 17, 23, 35, 6, 6, 37, 25, 19, 32, 13, 3482, 199, 161, 24335, 48, 7, 7, 50, 649, 182, 485, 89, 15, 151, 99, 530, 31, 29718, 63, 8, 8, 65, 48842, 33, 7775, 251, 3480, 17, 1068, 43, 26, 57799, 351, 53, 80, 9, 9, 82, 55, 378, 10405, 28, 197, 500, 19, 1574, 1151, 12151, 2143295, 39, 49, 5604, 99, 10, 10, 101, 227528

Offset: 1

Wolfdieter Lang, Nov 08 2002

If d(n)=A000037(n) is from A003654 (that is if the regular continued fraction for sqrt(d(n)) has odd (primitive) period length) then the -1 option applies. For such d(n) the minimal a(n) and b(n) numbers for the +1 option are 2*a(n)^2+1 and 2*a(n)*b(n), respectively (see Perron I, pp. 94,95).
If d(n)=A000037(n)= k^2+1, k=1,2,.., then the a^2 - d(n)*b^2 = -1 Pell equation has the minimal solution a(n)=k and b(n)=1. If d(n)=A000037(n)= k^2-1, k=2,3,..., then the a^2 - d(n)*b^2 = +1 Pell equation has the minimal solution a=k and b=1.
The general integer solutions (up to signs) of Pell equation a^2 - d(n)*b^2 = +1 with d(n)=A000037(n), but not from A003654, are a(n,p)= T(p+1,a(n)) and b(n,p)= b(n)*S(p,2*a(n)), p=0,1,... If d(n)=A000037(n) is also from A003654 then these solutions are a(n,p)= T(p+1,2*a(n)^2+1) and b(n,p)= 2*a(n)*b(n)*S(p,2*(2*a(n)^2+1)), p=0,1,... Here T(n,x), resp. S(n,x) := U(n,x/2), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310.
The general integer solutions (up to signs) of the Pell equation a^2 - d(n)*b^2 = -1 with d(n)=A000037(n)= A003654(k), for some k>=1, are a(n,p) = a(n)*(S(n,2*(2*a(n)^2)+1) + S(n-1,2*(2*a(n)^2)+1)) and b(n,p) = b(n)*(S(n,2*(2*a(n)^2)+1) - S(n-1,2*(2*a(n)^2)+1)) with the S(n,x) := U(n,x/2) Chebyshev polynomials. S(-1,x) := 0.
If the trivial solution x=1, y=0 is included, the sequence becomes A006702. - T. D. Noe, May 17 2007

			d=10=A000037(7)=A003654(3), therefore a(7)^2=10*b(7)^2 -1, i.e. 3^2=10*1^2 -1 and 2*a(7)^2+1=19 and 2*a(7)*b(7)=2*3*1=6 satisfy 19^2 - 10*6^2 = +1.
d=11=A000037(8) is not in A003654, therefore there is no (nontrivial) solution of the a^2 - d*b^2 = -1 Pell equation and a(8)=10 and b(8)=A077233(8)=3 satisfy 10^2 - 11*3^2 = +1.
10=d(7)=A000037(7)=A003654(3)=3^2+1 hence a(7)=3 and b(7)=1 are the smallest numbers satisfying a^2-10*b^2=-1.
8=d(6)=A000037(6)=3^2-1 (not in A003654) hence a(6)=3 and b(6)=1 are the smallest numbers satisfying a^2-8*b^2=+1.

T. Nagell, "Introduction to Number Theory", Chelsea Pub., New York, 1964, table p. 301.
O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 26, p. 91 with explanation on pp. 94,95).

Ray Chandler, Table of n, a(n) for n = 1..10000
A. M. Legendre, Fractions les plus simples m/n qui satisfont à l'équation m^2 - an^2 =+-1 pour tout nombre non quarré a depuis 2 jusqu'à 1003, Essai sur la Théorie des Nombres An VI, Table XII. [_Paul Curtz_, Apr 10 2019]
Index entries for sequences related to Chebyshev polynomials.

Cf. A000037, A003654, A003814, A006702, A033313, A077233.

nmax = 500;
nconv = 200; (* The number of convergents 'nconv' should be increased if the linear recurrence is not found for some terms. *)
nonSquare[n_] := n + Round[Sqrt[n]];
a[n_] := a[n] = Module[{lr}, lr = FindLinearRecurrence[ Numerator[ Convergents[ Sqrt[nonSquare[n]], nconv]]]; (1/2) SelectFirst[lr, #>1&]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Mar 10 2021 *)

a(n)=sqrt(A000037(n)*A077233(n)^2 + (-1)^(c(n))) with c(n)=1 if A000037(n)=A003654(k) for some k>=1 else c(n)=0.

A033313 Smallest positive integer x satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D and positive y.

Original entry on oeis.org

3, 2, 9, 5, 8, 3, 19, 10, 7, 649, 15, 4, 33, 17, 170, 9, 55, 197, 24, 5, 51, 26, 127, 9801, 11, 1520, 17, 23, 35, 6, 73, 37, 25, 19, 2049, 13, 3482, 199, 161, 24335, 48, 7, 99, 50, 649, 66249, 485, 89, 15, 151, 19603, 530, 31, 1766319049, 63, 8, 129, 65, 48842, 33

Offset: 1

Eric W. Weisstein

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
S. R. Finch, Class number theory [Cached copy, with permission of the author]
Bernard Frénicle de Bessy, Solutio duorum problematum circa numeros cubos et quadratos, (1657). Bibliothèque Nationale de Paris. See column C page 19.
H. W. Lenstra, jr., Solving the Pell Equation, Notices of the AMS, Vol.49, No.2, Feb. 2002, p.182-192.
F. Richman and R. Mines, Pell's equation
Derek Smith, Historical Overview of Pell Equations
Derek Smith, The Search For An Exhaustive Solution to Pell's Equation
Eric Weisstein's World of Mathematics, Pell Equation

See A033317 (for y's).

Cf. A000037, A002350, A077232, A077233.

F:= proc(d) local r,Q; uses numtheory;
  Q:= cfrac(sqrt(d),'periodic','quotients'):
  r:= nops(Q[2]);
  if r::odd then
    numer(cfrac([op(Q[1]),op(Q[2]),op(Q[2][1..-2])]))
  else
    numer(cfrac([op(Q[1]),op(Q[2][1..-2])]));
  fi
end proc:
map(F, remove(issqr,[$1..100])); # Robert Israel, May 17 2015

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[Sqrt[m]]; n = Length[Last[cf]]; If[n == 0, Return[{}]]; If[OddQ[n], n = 2n]; s = FromContinuedFraction[ContinuedFraction[Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
A033313 = DeleteCases[PellSolve /@ Range[100], {}][[All, 1]] (* Jean-François Alcover, Nov 21 2020, after N. J. A. Sloane in A002350 *)
Table[If[! IntegerQ[Sqrt[k]], {k,FindInstance[x^2 - k*y^2 == 1 && x > 0 && y > 0, {x, y},Integers]}, Nothing], {k, 2, 80}][[All, 2, 1, 1, 2]] (* Horst H. Manninger, Mar 28 2021 *)

a(n) = sqrt(1 + A000037(n)*A033317(n)^2), or

a(n) = sqrt(1 + (n + floor(1/2 + sqrt(n)))*A033317(n)^2). - Zak Seidov, Oct 24 2013

Offset switched to 1 by R. J. Mathar, Sep 21 2009

Name corrected by Wolfdieter Lang, Sep 03 2015

A033317 Smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D.

Original entry on oeis.org

2, 1, 4, 2, 3, 1, 6, 3, 2, 180, 4, 1, 8, 4, 39, 2, 12, 42, 5, 1, 10, 5, 24, 1820, 2, 273, 3, 4, 6, 1, 12, 6, 4, 3, 320, 2, 531, 30, 24, 3588, 7, 1, 14, 7, 90, 9100, 66, 12, 2, 20, 2574, 69, 4, 226153980, 8, 1, 16, 8, 5967, 4, 936, 30, 413, 2, 267000, 430, 3, 6630, 40, 6, 9

Offset: 1

Eric W. Weisstein

nonn

D = D(n) = A000037(n). - Wolfdieter Lang, Oct 04 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Laurent Beeckmans, Squares Expressible as Sum of Consecutive Squares, Am. Math. Monthly, Volume 101, Number 5, page 442, May 1994.
S. R. Finch, Class number theory [Cached copy, with permission of the author]
Bernard Frénicle de Bessy, Solutio duorum problematum circa numeros cubos et quadratos, (1657). Bibliothèque Nationale de Paris. See column B page 19.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Pell Equation

Cf. A000037, A033313 (for the x's), A077232, A077233.

F:= proc(d) local r,Q; uses numtheory;
  Q:= cfrac(sqrt(d),'periodic','quotients'):
  r:= nops(Q[2]);
  if r::odd then
    denom(cfrac([op(Q[1]),op(Q[2]),op(Q[2][1..-2])]))
  else
    denom(cfrac([op(Q[1]),op(Q[2][1..-2])]));
  fi
end proc:
map(F, remove(issqr,[$1..100])); # Robert Israel, May 17 2015

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[Sqrt[m]]; n = Length[Last[cf]]; If[n == 0, Return[{}]]; If[OddQ[n], n = 2n]; s = FromContinuedFraction[ContinuedFraction[Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
A033317 = DeleteCases[PellSolve /@ Range[100], {}][[All, 2]] (* Jean-François Alcover, Nov 21 2020, after N. J. A. Sloane in A002349 *)

a(n) = sqrt((A033313(n)^2 - 1)/A000037(n)). - Jinyuan Wang, Jul 09 2020

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

Original entry on oeis.org

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

N. J. A. Sloane

nonn

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).

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]

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

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 *)

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

cp's OEIS Frontend

A077232 a(n) is smallest natural number satisfying Pell equation a^2 - d(n)*b^2= +1 or = -1, with d(n)=A000037(n) (a nonsquare). Corresponding smallest b(n)=A077233(n).

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A033313 Smallest positive integer x satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D and positive y.

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A033317 Smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D.

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

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