A197170 -id:A197170 - cp's OEIS frontend

A197171 Values k such that singular quadratic unity of Q(k) have gcd(k,y) = 2.

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

Offset: 2

Artur Jasinski, Oct 11 2011

nonn

Conjecture: This sequence is infinite.

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

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

A197172 Values k such that singular quadratic unity of Q(k) have gcd(k, y) = 3.

Original entry on oeis.org

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

Offset: 2

Artur Jasinski, Oct 11 2011

nonn

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

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

A283395 Squarefree numbers m congruent to 1 modulo 4 such that the fundamental unit of the field Q(sqrt(m)) has the form x+y*sqrt(m) with x, y integers.

Original entry on oeis.org

17, 33, 37, 41, 57, 65, 73, 89, 97, 101, 105, 113, 129, 137, 141, 145, 161, 177, 185, 193, 197, 201, 209, 217, 233, 241, 249, 257, 265, 269, 273, 281, 305, 313, 321, 329, 337, 345, 349, 353, 373, 377, 381, 385, 389, 393, 401, 409, 417, 433, 449, 457, 465, 473, 481, 485, 489, 497, 505, 521, 537, 545, 553, 557, 561, 569, 573

Offset: 1

Emmanuel Vantieghem, Mar 07 2017

nonn

Squarefree integers m congruent to 1 modulo 4 such that the minimal solution of the Pell equation x^2 - d*y^2 = +-4 has both x and y even.
The sequence contains the squarefree numbers congruent to 5 modulo 8 that are not in A107997.
This sequence union A107997 = A039955.
This sequence contains all numbers of the form 4*k^2+1 (k > 1) that are squarefree.

			33 is in the sequence since the fundamental unit of the field Q(sqrt(33)) is 23+4*sqrt(33).
53 is not in the sequence since the fundamental unit of the field Q(sqrt(53)) is 3+omega, where omega = (1+sqrt(53))/2.

Z. I. Borevich and I. R. Shafarevich. Number Theory. Academic Press. 1966.

Keith Matthews, Finding the fundamental unit of a real quadratic field

Cf. A107997, A039955, A107998, A197170.

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A197171 Values k such that singular quadratic unity of Q(k) have gcd(k,y) = 2.

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A197172 Values k such that singular quadratic unity of Q(k) have gcd(k, y) = 3.

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Mathematica

A283395 Squarefree numbers m congruent to 1 modulo 4 such that the fundamental unit of the field Q(sqrt(m)) has the form x+y*sqrt(m) with x, y integers.

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