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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).
+20
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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
COMMENTS
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.
EXAMPLE
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.
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).
+20
2
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
COMMENTS
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)).
EXAMPLE
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.
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.
+10
11
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
COMMENTS
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.
EXAMPLE
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, 170], [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)], ...
The values D not a square that admit proper solutions (x, y) to the Pell equation x^2 - D*y^2 = +8, ordered increasingly.
+10
1
8, 17, 28, 41, 56, 73, 89, 92, 97, 113, 124, 136, 137, 161, 184, 188, 193, 217, 233, 241, 248, 281, 284, 313, 316, 329, 337, 353, 376, 392, 409, 412, 433, 449, 457, 476, 497, 508, 521, 553, 568, 569, 593, 601, 604, 617, 632, 641, 668, 673, 713, 721, 764, 769, 776, 796, 809, 824, 833, 857, 881, 889, 892, 929, 937, 952, 953, 956, 977, 1016, 1033, 1049, 1052, 1057, 1081, 1084, 1097, 1148, 1153
COMMENTS
This sequence is the union of A263012 and A264354. See these sequences for details and examples. For square D the only solution is D=1 with (3, 1).
Improper solutions derive from 2*(X, Y) with (X, Y) proper solutions for D a member of A261246.
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