A031397 -id:A031397 - cp's OEIS frontend

A003654 Squarefree integers m such that the fundamental unit of Q(sqrt(m)) has norm -1. Also, squarefree integers m such that the Pell equation x^2 - m*y^2 = -1 is soluble.

2, 5, 10, 13, 17, 26, 29, 37, 41, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 130, 137, 145, 149, 157, 170, 173, 181, 185, 193, 197, 202, 218, 226, 229, 233, 241, 257, 265, 269, 274, 277, 281, 290, 293, 298, 313, 314, 317, 337, 346, 349, 353, 362

Offset: 1

N. J. A. Sloane, Mira Bernstein. Entry revised by N. J. A. Sloane, Jun 11 2012

nonn

The squarefree elements of A003814 and A172000. - Max Alekseyev, Jun 01 2009
Together with {1} and A031398 forms a disjoint partition of A020893. That is, A020893 = {1} U A003654 U A031398. - Max Alekseyev, Mar 09 2010
Squarefree integers m such that Q(sqrt(m)) contains the infinite continued fraction [k, k, k, k, k, ...] for some positive integer k. For example, Q(sqrt(5)) contains [1, 1, 1, 1, 1, ...] which equals (1 + sqrt(5))/2. - Greg Dresden, Jul 23 2010

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 46.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 56.
W. Paulsen, Calkin-Wilf sequences for irrational numbers, Fib. Q., 61:1 (2023), 51-59.
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).

R. J. Mathar, Table of n, a(n) for n = 1..9446
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
P. Seeling, Ueber die Aufloesung der Gleichung x^2-Ay^2=+-1 in ganzen Zahlen, wo A positiv und kein vollstaendiges Quadrat sein muss, Archiv der Mathematik und Physik, Vol. 52 (1871), p. 40-49.

Cf. A031396, A031397, A003814, A249021.

isA003654 := proc(n)
    local cf,p ;
    if not numtheory[issqrfree](n) then
        return false;
    end if;
    for p in numtheory[factorset](n) do
        if modp(p,4) = 3 then
            return false;
        end if;
    end do:
    cf := numtheory[cfrac](sqrt(n),'periodic','quotients') ;
    type( nops(op(2,cf)),'odd') ;
end proc:
A003654 := proc(n)
    option remember;
    local a;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA003654(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A003654(n),n=1..40) ; # R. J. Mathar, Oct 19 2014

Reap[For[n = 2, n < 1000, n++, If[SquareFreeQ[n], sol = Solve[x^2 - n y^2 == -1, {x, y}, Integers]; If[sol != {}, Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Mar 24 2020 *)

Edited by Max Alekseyev, Mar 17 2010

A031398 Squarefree n with no 4k+3 factors such that Pell equation x^2 - n y^2 = -1 is insoluble.

Original entry on oeis.org

34, 146, 178, 194, 205, 221, 305, 377, 386, 410, 466, 482, 505, 514, 545, 562, 674, 689, 706, 745, 793, 802, 866, 890, 898, 905, 1154, 1186, 1202, 1205, 1234, 1282, 1345, 1346, 1394, 1405, 1469, 1513, 1517, 1537, 1538, 1717, 1762, 1802, 1858

Offset: 1

David W. Wilson

nonn

Or, numbers n which are the sum of two relatively-prime squares but for which x^2 - n*y^2 does not represent -1.

Together with {1} and A003654 forms a disjoint partition of A020893. That is, A020893 = {1} U A003654 U A031398. - Max Alekseyev, Mar 09 2010

Harvey Cohn, "Advanced Number Theory".

Jean-François Alcover, Table of n, a(n) for n = 1..1000
Janis Kuzmanis, A simple solvability criterion for the negative Pell equation, hal-02502164, Mathematics [math] / Number Theory [math.NT], (2020).
K. Lakshmi, R. Someshwari, On The Negative Pell Equation y^2 = 72x^2 - 23, International Journal of Emerging Technologies in Engineering Research (IJETER), Volume 4, Issue 7, July (2016).
J. P. Robertson and K. R. Matthews, A continued fraction approach to a result of Feit, Amer. Math. Monthly, 115 (No. 4, 2008), 346-349.
R. Suganya, D. Maheswari, On the Negative Pellian Equation y^2 = 110 * x^2 - 29, Journal of Mathematics and Informatics, Vol. 11 (2017), pp. 63-71.
S. Vidhyalakshmi, V. Krithika, K. Agalya, On The Negative Pell Equation y^2 = 72x^2 - 8, International Journal of Emerging Technologies in Engineering Research (IJETER) 4:2 (2016).

Cf. A031396, A031397.

sel = Select[Range[2000], SquareFreeQ[#] && FreeQ[Mod[FactorInteger[#][[All, 1]], 4], 3]&]; r[n_] := Reduce[x^2-n*y^2 == -1, {x, y}, Integers]; Reap[For[n=1, n <= Length[sel], n++, an = sel[[n]]; If[r[an] === False, Print[an]; Sow[an]]]][[2, 1]] (* Jean-François Alcover, Feb 04 2014 *)

Edited by N. J. A. Sloane, Apr 28 2008, at the suggestion of Artur Jasinski

A031399 Numbers n with no 4k+3 factors such that Pell equation x^2 - n y^2 = -1 insoluble.

Original entry on oeis.org

4, 8, 16, 20, 25, 32, 34, 40, 52, 64, 68, 80, 100, 104, 116, 128, 136, 146, 148, 160, 164, 169, 178, 194, 200, 205, 208, 212, 221, 232, 244, 256, 260, 272, 289, 292, 296, 305, 320, 328, 340, 356, 377, 386, 388, 400, 404, 410, 416, 424, 436, 452

Offset: 1

David W. Wilson

nonn

"Advanced Number Theory" by Harvey Cohn.

Joerg Arndt, Matters Computational (The Fxtbook)

Cf. A031396, A031397, A031398.

cp's OEIS Frontend

A003654 Squarefree integers m such that the fundamental unit of Q(sqrt(m)) has norm -1. Also, squarefree integers m such that the Pell equation x^2 - m*y^2 = -1 is soluble.

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A031398 Squarefree n with no 4k+3 factors such that Pell equation x^2 - n y^2 = -1 is insoluble.

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Mathematica

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A031399 Numbers n with no 4k+3 factors such that Pell equation x^2 - n y^2 = -1 insoluble.

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Author

Keywords

References

Links

Crossrefs