A194366 -id:A194366

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.

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This sequence is a subsequence of A172000.

LINKS

Table of n, a(n) for n=1..56.

MATHEMATICA

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

CROSSREFS

Cf. A087643, A172000, A194366.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 10 2011

EXTENSIONS

Definition clarified by Emmanuel Vantieghem, Mar 06 2017

STATUS

approved

A197127

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

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

LINKS

Table of n, a(n) for n=1..56.

EXAMPLE

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

MATHEMATICA

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*)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 10 2011

STATUS

approved

A197128

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

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

LINKS

Table of n, a(n) for n=1..66.

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 10 2011

STATUS

approved

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.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

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

LINKS

Table of n, a(n) for n=2..46.

FORMULA

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

EXAMPLE

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).

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 11 2011

STATUS

approved

A197169

Values of gcd(n,y) for successive y = A197128(n).

+10
1

2, 2, 2, 2, 2, 2, 2, 46, 2, 2, 2, 2, 3, 10, 6, 2, 3, 3, 2, 2, 2, 6, 5, 2, 2, 5, 2, 2, 2, 2, 2, 2, 22, 5, 2, 2, 3, 2, 2, 3, 2, 2, 3, 46, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 14, 2, 4, 3, 2, 2, 2, 4, 14, 3, 2, 3, 2, 5, 2, 2, 2, 5, 2, 2, 2, 3, 6, 29, 3, 2, 2, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..84.

MATHEMATICA

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, GCD[d4, n]]]], {n, 2, 20000}]; cr

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 11 2011

STATUS

approved

A197171

Values k such that singular quadratic unity of Q(k) have gcd(k,y) = 2.

+10
1

6, 14, 22, 30, 34, 38, 42, 54, 56, 62, 66, 86, 94, 102, 110, 118, 126, 132, 134, 138, 142, 146, 150, 156, 158, 166, 174, 178, 182, 186, 190, 194, 198, 206, 210, 214, 220, 222, 228, 230, 246, 254, 258, 262, 270, 278, 282, 286, 294, 302, 306, 310, 322, 326

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

Conjecture: This sequence is infinite.

LINKS

Table of n, a(n) for n=2..55.

MATHEMATICA

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[If[ck[[n]] == 2, AppendTo[aa, cr[[n]]]], {n, 1, Length[cr]}]; aa

CROSSREFS

Cf. A087643, A172000, A194366, A197115, A197127, A197128, A197170.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 11 2011

STATUS

approved

A197120

Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(d))has norm -1 and minimum one from two parts of fundamental unit are not integer.

+10
0

5, 13, 20, 29, 45, 52, 53, 61, 80, 85, 109, 116, 117, 125, 149, 157, 173, 180, 181, 208, 212, 229, 244, 245, 261, 277, 293, 317, 320, 325, 340, 365, 397, 405, 421, 436, 445, 461, 464, 468, 477, 493, 500, 509, 533, 541, 549, 565, 596, 605, 613, 628, 629, 637

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers which occured in A172000 and not in A197115.

LINKS

Table of n, a(n) for n=1..54.

MATHEMATICA

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, 2000}]; cr

CROSSREFS

Cf. A087643, A172000, A194366, A197115.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 10 2011

STATUS

approved

A197121

Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(d))has norm 1 and minimum one from two parts of fundamental unit are not integer.

+10
0

21, 69, 77, 84, 93, 133, 165, 189, 205, 213, 221, 237, 253, 276, 285, 301, 308, 309, 336, 341, 357, 372, 413, 429, 437, 453, 469, 501, 517, 525, 532, 581, 589, 597, 621, 645, 660, 669, 693, 717, 741, 749, 756, 789, 805, 820, 837, 852, 861, 869, 884, 893, 917

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers which occured in A087643 and not in A194366.

LINKS

Table of n, a(n) for n=1..53.

MATHEMATICA

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, 2000}]; cr

CROSSREFS

Cf. A087643, A172000, A194366, A197115.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 10 2011

STATUS

approved

A197172

Values k such that singular quadratic unity of Q(k) have gcd(k, y) = 3.

+10
0

69, 87, 93, 159, 177, 183, 249, 267, 276, 312, 321, 327, 348, 372, 387, 417, 471, 597, 633, 636, 699, 711, 717, 723, 741, 747, 831, 849, 879, 921, 927, 987, 993, 1005, 1068, 1104, 1137, 1179, 1248, 1251, 1272, 1293, 1299, 1317, 1320, 1353, 1359, 1383, 1392

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

LINKS

Table of n, a(n) for n=2..50.

MATHEMATICA

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[If[ck[[n]] == 3, AppendTo[aa, cr[[n]]]], {n, 1, Length[cr]}]; aa

CROSSREFS

Cf. A087643, A172000, A194366, A197115, A197127, A197128, A197170, A197171.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 11 2011

STATUS

approved