A264351 -id:A264351 - cp's OEIS frontend

A263012 Odd numbers D not a square that admit proper solutions (x, y) to the Pell equation x^2 - D*y^2 = +8 with both x and y odd.

17, 41, 73, 89, 97, 113, 137, 161, 193, 217, 233, 241, 281, 313, 329, 337, 353, 409, 433, 449, 457, 497, 521, 553, 569, 593, 601, 617, 641, 673, 713, 721, 769, 809, 833, 857, 881, 889, 929, 937, 953, 977, 1033, 1049, 1057, 1081, 1097, 1153, 1169, 1193, 1201, 1217, 1241, 1249, 1289, 1321, 1337, 1361, 1409, 1433, 1457, 1481, 1513, 1553, 1561, 1609, 1633, 1649, 1657, 1673, 1697, 1721, 1753, 1777, 1801, 1817, 1841, 1873, 1889, 1913, 1921, 1993

Offset: 1

Wolfdieter Lang, Nov 17 2015

nonn

These are the nonsquare odd numbers D that admit proper solutions (x, y) to the generalized Pell equation x^2 - D*y^2 = +8 with both x and y odd. They are given by D == 1 (mod 8), not a square, no prime factors 3 or 5 (mod 8) in the composite case (see A263011), and they are not exceptional values which are given in A264348. Up to the number 2000 these exceptional values are 257, 401, 577, 697, 761, 1009, 1129, 1297, 1393, 1489, 1601, 1897. [sequence reference corrected by Peter Munn, Jun 19 2020]
The corresponding positive proper fundamental solutions (x1(D), y1(D)) for the first class are given in A264349 and A264350. There always seem to be two conjugacy classes. The positive proper fundamental solution of the second class (x2, y2) is, for given D, obtained by applying the matrix M(D) = matrix[[x0(D), D*y0(D)],[y0(D), x0(D)]] on (x1(D), -y1(D))^T (T for transposed). Here (x0(D), y0(D)) is the positive fundamental solution of the Pell equation x^2 - D*y^2 = +1 (which is always proper). See the appropriate entries of A033313 and A033317 for these solutions. There would be only one class (the ambiguous case) if this application of M(D) would lead to (x1(D), y1(D))^T. This does not seem to happen. The positive proper fundamental solutions (x2(D), y2(D)) of the second class are given in A264351 and A264353.
The case of odd D with both y and x even leads to improper solutions obtained from the +2 Pell equation (see A261246), e.g., D = 7 has the fundamental positive improper solution (6, 2) = 2*(3, 1) obtained from the proper solution (3, 1) of x^2 - 7*y^2 = +2 (see A261247(2) and A261248(2)). There is only one class of solutions (ambiguous case).
The case of even D with y odd and x even needs D == 0 (mod 4). See 4*A261246 = A264354 for the even D values that admit proper solutions. There appear one or two classes of solutions in this case.
The improper solutions with even D and both x and y even, come from X^2 - D*Y2 = +2 which needs D/2 odd without prime factors 3 or 5 (mod 8) in the composite case. Such D values that do not admit a solution are called exceptional and are given by A264352.
This is a proper subsequence of A263011.

			The first positive fundamental solutions of the first class (x1(n), y1(n)) are (the first entry gives D(n) = a(n)):
  [17, (5, 1)], [41, (7, 1)], [73, (9, 1)],
  [89, (217, 23)], [97, (69, 7)], [113, (11, 1)], [137, (199, 17)], [161, (13, 1)],
  [193, (56445, 4063)], [217, (15, 1)],
  [233, (6121, 401)], [241, (46557, 2999)],
  [281, (17, 1)], [313, (9567711, 540799)],
  [329, (127, 7)], [337, (73829571, 4021753)], ...
The first positive fundamental solutions of the second class (x2(n), y2(n)) are:
  [17, (29, 7)], [41, (1223, 191)],
  [73, (1040241, 121751)], [89, (9217, 977)],
  [97, (3642669, 369857)], [113, (445435, 41903)], [137, (122279, 10447)], [161, (3667, 289)],
  [193, (441089445, 31750313)],
  [217, (1034361, 70217)], [233, (700801, 45911)], [241, (866477098293, 55814696449)], ...

Cf. A261246, A263011, A264348, A264349, A264350, A264351, A264353, A264354.

A264349 The x member of the positive proper fundamental solution (x = x1(n), y = y1(n)) of the first class for the Pell equation x^2 - D(n)*y^2 = +8 for odd D(n) = A263012(n).

Original entry on oeis.org

5, 7, 9, 217, 69, 11, 199, 13, 56445, 15, 6121, 46557, 17, 9567711, 127, 73829571, 19, 101080863, 21, 35927573, 1582431, 379, 23, 729, 167, 32899, 689581594497, 25, 1348157, 207495429, 1255, 27, 146613, 278601383, 29, 21751, 4777049027, 33814353, 19022653, 5062449, 31, 43541, 225

Offset: 1

Wolfdieter Lang, Nov 17 2015

nonn

The corresponding y = y1(n) member is given by A264350(n).

See A263012 for details and examples.

			n=1:  5^2 - 17*1^2  = 8. n=4: 217^2 - 89*23^2  = 8.

Cf. A263011, A263012, A264350, A264351, A264353.

a(n)^2 - D(n)*y1(n)^2 = 8 with D(n) = A263012(n) and y1(n) = A264350(n), n >= 1.

A264350 The y member of the positive proper fundamental solution (x = x1(n), y = y1(n)) of the first class for the Pell equation x^2 - D(n)*y^2 = +8 for odd D(n) = A263012(n).

Original entry on oeis.org

1, 1, 1, 23, 7, 1, 17, 1, 4063, 1, 401, 2999, 1, 540799, 7, 4021753, 1, 4998127, 1, 1695527, 74023, 17, 1, 31, 7, 1351, 28128619951, 1, 53249, 7998361, 47, 1, 5287, 9795103, 1, 743, 160942871, 1134097, 624113, 165383, 1, 1393, 7

Offset: 1

Wolfdieter Lang, Nov 17 2015

nonn

The corresponding x = x1(n) member is given by A264349.

See A263012 for details and examples.

			n=5: 69^2  - 97*7^2 = 8.

Cf. A263011, A263012, A264349, A264351, A264353.

x1(n)^2 - D(n)*a(n)^2 = 8 with D(n) = A263012(n) and x1(n) = A264349(n), n >= 1.

A264353 The y member of the positive proper fundamental solution (x = x2(n), y = y2(n)) of the second class for the Pell equation x^2 - D(n)*y^2 = +8 for odd D(n) = A263012(n).

Original entry on oeis.org

7, 191, 121751, 977, 369857, 41903, 10447, 289, 31750313, 70217, 45911, 55814696449, 31976153111, 760633409, 4127, 6090942263, 114449983, 49021742073857, 961536627246198743, 377247137, 825722983360793, 569, 252098751014159, 145746599, 16830802006252297, 3602441, 9204879914229161, 21735887654887, 305798976086839, 3541220927745592511, 631, 103083370004609, 527123470001650790344559393, 190027177610417, 45407

Offset: 1

Wolfdieter Lang, Nov 17 2015

nonn

The corresponding x = x2(n) member is given by A264351(n).

See A263012 for details and examples.

			n=5: 3642669^2  - 97*369857^2 = 8.

Cf. A263011, A263012, A264349, A264350, A264351.

x2(n)^2 - D(n)*a(n)^2 = 8 with D(n) = A263012(n) and x2(n) = A264351(n), n >= 1.

A264438 One-half of the x member of the positive proper fundamental solution (x = x2(n), y = y2(n)) of the second class for the Pell equation x^2 - D(n)*y^2 = +8 for even D(n) = A264354(n).

Original entry on oeis.org

2, 45, 4, 235, 118521, 6, 156, 665, 8, 410581, 1431, 1464, 10, 217061235, 2629, 20578212225, 12, 143681684300109, 88, 4355, 53946009001, 14, 4149148875801021, 244, 6705, 108, 30839304871, 16, 103789115, 78990793279586649, 9775, 2068, 138751721731, 18, 7987764, 2984191388685, 13661, 5246209297401255, 406200, 5142295

Offset: 1

Wolfdieter Lang, Nov 19 2015

nonn

The corresponding y2(n) value is given by A264439(n). The positive fundamental solution (x1(n), y1(n)) of the first class is given by (2*A261247(n), A261248(n)).
There is only one class of proper solutions for those D = D(n) = A264354(n) that lead to (x1(n), y1(n)) = (x2(n), y2(n)).
See A264354 for comments and examples.

			n=2: D(2) = 28, (2*45)^2 - 28*17^2 = +8. The first class solution was (2*3)^2 - 28*1^2 = +8. This is a D case with two classes of proper solutions.
n=3: D(3) = 56, (2*4)^2 - 56*1^2 = +8. The first class has the same solution, therefore this D has only one class of proper solutions.

Cf. A264354, A261247, A261248, A264439, A263012 (odd D), A264349, A264350, A264351, A264353.

A264439 The y member of the positive proper fundamental solution (x = x2(n), y = y2(n)) of the second class for the Pell equation x^2 - D(n)*y^2 = +8 for even D(n) = A264354(n).

Original entry on oeis.org

1, 17, 1, 49, 21287, 1, 23, 97, 1, 48727, 161, 151, 1, 21387679, 241, 1826021057, 1, 11692649642023, 7, 337, 3903396217, 1, 294125365483681, 17, 449, 7, 1994828801, 1, 6399911, 4798348971487087, 577, 119, 7867888313, 1, 437071, 161131189369, 721, 273849896195263, 20783, 262759

Offset: 1

Wolfdieter Lang, Nov 19 2015

nonn

The corresponding x2(n) value is given by A264438(n). The positive fundamental solution (x1(n), y1(n)) of the first class is given by (2*A261247(n), A261248(n)).

There is only one class of proper solutions for those D = D(n) = A264354(n) values leading to (x1(n), y1(n)) = (x2(n), y2(n)).

			n=1: D(1) = 8, (2*2)^2 - 8*1^2 = +8. The first class positive fundamental solution was identical, thus there is only one class of proper solutions for D = 8.
n=5: D(5) = 124, (2*118521)^2 - 124*21287^2 = +8. The first class solution was (2*39)^2 - 124*7^2 = +8. Thus there are two classes, conjugated to each other for this D value.

Cf. A264354, A261247 (x1/2), A261248 (y1), A264438 (x2/2), A263012 (odd D), A264349, A264350, A264351, A264353.

cp's OEIS Frontend

A263012 Odd numbers D not a square that admit proper solutions (x, y) to the Pell equation x^2 - D*y^2 = +8 with both x and y odd.

Views

Author

Keywords

Comments

Examples

Crossrefs

A264349 The x member of the positive proper fundamental solution (x = x1(n), y = y1(n)) of the first class for the Pell equation x^2 - D(n)*y^2 = +8 for odd D(n) = A263012(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A264350 The y member of the positive proper fundamental solution (x = x1(n), y = y1(n)) of the first class for the Pell equation x^2 - D(n)*y^2 = +8 for odd D(n) = A263012(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A264353 The y member of the positive proper fundamental solution (x = x2(n), y = y2(n)) of the second class for the Pell equation x^2 - D(n)*y^2 = +8 for odd D(n) = A263012(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A264438 One-half of the x member of the positive proper fundamental solution (x = x2(n), y = y2(n)) of the second class for the Pell equation x^2 - D(n)*y^2 = +8 for even D(n) = A264354(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

A264439 The y member of the positive proper fundamental solution (x = x2(n), y = y2(n)) of the second class for the Pell equation x^2 - D(n)*y^2 = +8 for even D(n) = A264354(n).

Views

Author

Keywords

Comments

Examples

Crossrefs