A172000 -id:A172000 - cp's OEIS frontend

A087643 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n)) has norm 1.

3, 6, 7, 11, 12, 14, 15, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 34, 35, 38, 39, 42, 43, 44, 46, 47, 48, 51, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 69, 70, 71, 75, 76, 77, 78, 79, 83, 84, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 102, 103, 105, 107, 108, 110, 111, 112, 114

Offset: 3

Thomas Baruchel, Sep 16 2003

No quadratic number with a fully periodical continued fraction of period 1 can be written as (a+b*sqrt(n))/c with n allowed to have square factors.

Subsequence of the squarefree terms is A087642.

Cf. A087642, A172000.

cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d3 = Expand[(d1 + d2) (d1 - d2)]; If[d3 == 1, AppendTo[cr, n]]], {n, 2, 1000}]; cr (* Artur Jasinski, Oct 10 2011 *)

Edited by Max Alekseyev, May 03 2010

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.

Original entry on oeis.org

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

A077426 Nonsquare integers n such that the continued fraction (sqrt(n)+1)/2 has odd period length.

Original entry on oeis.org

5, 13, 17, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 173, 181, 185, 193, 197, 229, 233, 241, 257, 265, 269, 277, 281, 293, 313, 317, 325, 337, 349, 353, 365, 373, 389, 397, 401, 409, 421, 425, 433, 445, 449, 457, 461, 481, 485

Offset: 1

Wolfdieter Lang, Nov 29 2002

Nonsquare integers n for which Pell equation x^2 - n*y^2 = -4 has infinitely many integer solutions. The smallest solutions are given in A078356 and A078357.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, table p. 108).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wikipedia, Pell's equation

A subsequence of A077425.
Odd elements of A003814.
Cf. A077427, A172000.

isOddPrim := proc(n::integer)
    local cf;
    cf := numtheory[cfrac]((sqrt(n)+1)/2,'periodic','quotients') ;
    if nops(op(2,cf)) mod 2 =1 then
        RETURN(true) ;
    else
        RETURN(false) ;
    fi ;
end:
notA077426 := proc(n::integer)
    if issqr(n) then
        RETURN(true) ;
    elif not isOddPrim(n) then
        RETURN(true) ;
    else
        RETURN(false) ;
    fi ;
end:
A077426 := proc(n::integer)
    local resul,i ;
    resul := 5 ;
    i := 1 ;
    while i < n do
        resul := resul+4 ;
        while notA077426(resul) do
            resul := resul+4 ;
        od ;
        i:= i+1 ;
    od ;
    RETURN(resul) ;
end:
for n from 1 to 61 do print(A077426(n)) ; od : # R. J. Mathar, Apr 25 2006

fQ[n_] := !IntegerQ@ Sqrt@ n && OddQ@ Length@ ContinuedFraction[(Sqrt@ n + 1)/2][[2]]; Select[Range@ 500, fQ] (* Robert G. Wilson v, Nov 17 2012 *)

Edited and extended by Max Alekseyev, Mar 03 2010

Edited by Max Alekseyev, Mar 05 2010

A194366 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n)) has norm 1 and can be written as x+y*sqrt(d) with integers x, y where d is the squarefree part of n.

Original entry on oeis.org

3, 6, 7, 11, 12, 14, 15, 19, 22, 23, 24, 27, 28, 30, 31, 33, 34, 35, 38, 39, 42, 43, 44, 46, 47, 48, 51, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 70, 71, 75, 76, 78, 79, 83, 86, 87, 88, 91, 92, 94, 95, 96, 99, 102, 103, 105, 107, 108, 110, 111, 112, 114, 115

Offset: 1

Artur Jasinski, Oct 10 2011

nonn

This sequence is a subsequence of A087643.

			35 belongs to this sequence because x^2 + 35*y^2 = 1 has the integer solution x=6, y=1.

Cf. A087643, A172000.

cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d3 = Expand[(d1 + d2) (d1 - d2)]; If[d3 == 1, k1 = Max[Denominator[d1], Denominator[d2]];  If[k1 == 1, AppendTo[cr, n]]]], {n, 2, 100}]; cr

Definition clarified by Emmanuel Vantieghem, Mar 06 2017

A197115 Nonsquare positive integers k such that the fundamental unit of the quadratic field Q(sqrt(k)) has norm -1 and can be written as x + y*sqrt(d) with integers x, y where d is the squarefree part of k.

Original entry on oeis.org

2, 8, 10, 17, 18, 26, 32, 37, 40, 41, 50, 58, 65, 68, 72, 73, 74, 82, 89, 90, 97, 98, 101, 104, 106, 113, 122, 128, 130, 137, 145, 148, 153, 160, 162, 164, 170, 185, 193, 197, 200, 202, 218, 226, 232, 233, 234, 241, 242, 250, 257, 260, 265, 269, 272, 274

Offset: 1

Artur Jasinski, Oct 10 2011

nonn

This sequence is a subsequence of A172000.

Cf. A087643, A172000, A194366.

cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d3 = Expand[(d1 + d2) (d1 - d2)]; If[d3 == -1, k1 = Max[Denominator[d1], Denominator[d2]];   If[k1 == 1, AppendTo[cr, n]]]], {n, 2, 400}]; cr

Definition clarified by Emmanuel Vantieghem, Mar 06 2017

A197127 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(d))is singular.

Original entry on oeis.org

6, 14, 22, 30, 34, 38, 42, 46, 54, 56, 62, 66, 69, 70, 78, 86, 87, 93, 94, 102, 110, 114, 115, 118, 126, 130, 132, 134, 138, 142, 146, 150, 154, 155, 156, 158, 159, 166, 174, 177, 178, 182, 183, 184, 185, 186, 190, 194, 198, 206, 210, 214, 220, 222, 228, 230

Offset: 1

Artur Jasinski, Oct 10 2011

nonn

x^2+n*y^2=(+/-)2^s where s is 0 or 1.

Definition: Unity is singular when GCD[n,y]<>1.

			a(1)=6 because unity of quadratic field  Q(6) is 5+2*Sqrt[6] and GCD[2,6]=2 <>1.

Cf. A087643, A172000, A194366, A197115, A197128.

cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, , AppendTo[cr, n]]], {n, 2, 330}]; cr (*Artur Jasinski*)

A197128 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n))is not singular.

Original entry on oeis.org

2, 3, 5, 7, 8, 10, 11, 12, 13, 15, 17, 18, 19, 20, 21, 23, 24, 26, 27, 28, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 63, 65, 67, 68, 71, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 88, 89, 90, 91, 92

Offset: 1

Artur Jasinski, Oct 10 2011

nonn

x^2+n*y^2=(+/-)2^s where s is 0 or 1.

Definition: Unity is singular when GCD[n,y]<>1.

Cf. A087643, A172000, A194366, A197115, A197127.

cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, AppendTo[cr, n]]], {n, 2, 330}]; cr

A197170 Smallest k such that the fundamental unit (x+yw) or (x+yw)/2 of the real quadratic field Q(sqrt(k)) obeys gcd(k,y)=n.

Original entry on oeis.org

6, 69, 248, 115, 78, 511, 1016, 603, 70, 385, 3432, 793, 238, 2655, 14224, 1241, 3186, 703, 3980, 9177, 154, 736, 456, 1825, 3172, 13959, 2884, 319, 1110, 4619, 7136, 10659, 7174, 10255, 44856, 7067, 2926, 16185, 54280, 779, 7602, 10879, 22088, 10215, 46

Offset: 2

Artur Jasinski, Oct 11 2011

nonn

Conjecture: For every n such a quadratic field with minimum k exists.

			For n=2 the unit is 2*w-5 with k=6.
For n=3 the unit is (3*w+25)/2 with k=69.
For n=4 the unit is (4*w-63) with k=248.
For n=5 the unit is 105*w-1126 with k=115.
For n=7 the unit is 185290497*w-4188548960 with k=511 (and this x and y appear in A041976 and A041977).

Cf. A087643, A172000, A194366, A197115, A197127, A197128.

cr = {}; ck = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, , AppendTo[ck, GCD[d4, n]];  AppendTo[cr, n]]], {n, 2, 200000}]; aa = {}; Do[AppendTo[aa, cr[[First[Position[ck, n]][[1]]]]], {n, 2, 99}]; aa

k = A197127(m) where m is the smallest m such that A197169(m)=n.

A172001 Nonsquare positive integers n such that Pell equation y^2 - n*x^2 = -1 has rational solutions but the norm of fundamental unit of quadratic field Q(sqrt(n)) is 1.

Original entry on oeis.org

34, 136, 146, 178, 194, 205, 221, 305, 306, 377, 386, 410, 466, 482, 505, 514, 544, 545, 562, 584, 674, 689, 706, 712, 745, 776, 793, 802, 820, 850, 866, 884, 890, 898, 905, 1154, 1186, 1202, 1205, 1220, 1224, 1234, 1282, 1314, 1345, 1346, 1394, 1405, 1469

Offset: 1

Max Alekseyev, Jan 21 2010

nonn

If the fundamental unit y0 + x0*sqrt(n) of Q(sqrt(n)) has norm -1, then (x0,y0) represents a rational solution to Pell equation y^2 - n*x^2 = -1. For n in this sequence, rational solutions exist but not delivered by the fundamental unit.

Set difference of A000415 and its subsequence A172000.
Set difference of A087643 and its subsequence A022544.
Squarefree terms form A031398.
Odd terms form A249052.
Cf. A031399, A137351.

A positive integer n is in this sequence iff its squarefree core A007913(n) belongs to A031398.

Edited by Max Alekseyev, Mar 09 2010

A227939 Values of n such that the equation x^2 - 2ny^2 = n has integer solutions.

Original entry on oeis.org

1, 3, 4, 9, 11, 12, 16, 19, 25, 27, 33, 36, 43, 44, 48, 49, 51, 57, 59, 64, 67, 73, 75, 76, 81, 83, 89, 99, 100, 107, 108, 121, 123, 129, 131, 132, 139, 144, 147, 163, 169, 171, 172, 176, 177, 179, 187, 192

Offset: 1

Colin Barker, Oct 07 2013

nonn

			59 appears in the sequence because the equation x^2 - 118*y^2 = 59 has integer solutions.

Vincenzo Librandi, Table of n, a(n) for n = 1..440

Cf. A172000, A229848

Select[Range[200],Length[FullSimplify[Solve[x^2-2*#*y^2==#,{x,y},Integers]/.C[1]->1]]>0&] (* Vaclav Kotesovec, Oct 08 2013 *)

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A087643 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n)) has norm 1.

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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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A077426 Nonsquare integers n such that the continued fraction (sqrt(n)+1)/2 has odd period length.

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A194366 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n)) has norm 1 and can be written as x+y*sqrt(d) with integers x, y where d is the squarefree part of n.

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A197115 Nonsquare positive integers k such that the fundamental unit of the quadratic field Q(sqrt(k)) has norm -1 and can be written as x + y*sqrt(d) with integers x, y where d is the squarefree part of k.

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A197127 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(d))is singular.

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A197128 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n))is not singular.

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A197170 Smallest k such that the fundamental unit (x+y*w) or (x+y*w)/2 of the real quadratic field Q(sqrt(k)) obeys gcd(k,y)=n.

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A197170 Smallest k such that the fundamental unit (x+yw) or (x+yw)/2 of the real quadratic field Q(sqrt(k)) obeys gcd(k,y)=n.

A227939 Values of n such that the equation x^2 - 2ny^2 = n has integer solutions.