A263012 -id:A263012 - cp's OEIS frontend

A264351 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 odd D(n) = A263012(n).

29, 1223, 1040241, 9217, 3642669, 445435, 122279, 3667, 441089445, 1034361, 700801, 866477098293, 536018000953, 13456978719, 74857, 111814836531, 2150319955, 991403378623407, 20008281106509024285, 7993723517, 17651941238270751, 12685, 5754260987908903, 3427370031, 401477253306854423, 87725171, 225660404935077273, 539908248773809, 7742211688546213, 91867215744119283021, 16849, 2767937250852453, 14617581481552992324155240733, 5404928819007703, 1310525

Offset: 1

Wolfdieter Lang, Nov 17 2015

nonn

The corresponding y = y2(n) member are given by A264350(n).

See A263012 for details and examples.

			n=1:  29^2 - 17*7^2  = 8. n=4:  9217^2 - 89*977^2  = 8.

Cf. A263011, A263012, A264349, A264350, A264353.

a(n)^2 - D(n)*y2(n)^2 = 8 with D(n) = A263012(n) and y2(n) = A264351(n), n >= 1.

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.

A263011 Numbers D == 1 (mod 8), not a square, and if composite without prime factors 3 or 5 (mod 8).

Original entry on oeis.org

17, 41, 73, 89, 97, 113, 137, 161, 193, 217, 233, 241, 257, 281, 313, 329, 337, 353, 401, 409, 433, 449, 457, 497, 521, 553, 569, 577, 593, 601, 617, 641, 673, 697, 713, 721, 761, 769, 809, 833, 857, 881, 889, 929, 937, 953, 977, 1009, 1033, 1049, 1057, 1081, 1097, 1129, 1153, 1169, 1193, 1201, 1217

Offset: 1

Wolfdieter Lang, Nov 17 2015

These numbers are the odd D candidates for the (generalized) Pell equation x^2 - D*y^2 = +8 which could have proper solutions (x, y) with x and y both odd (and gcd(x, y) = 1).
Proof: Put x =2*X + 1, y = 2*Y + 1; then 8*(T(X) - D*T(Y)) = 8 - 1 + D = 7 + D, with the triangular numbers T = A000217. Hence, D == -7 (mod 8) == +1 (mod 8). Only nonsquare numbers D are considered for the Pell equation (square D leads to a factorization with only one solution: D = 1, (x, y) = (3, 1)). Consider a prime factor p == 3 or 5 (mod 8) (A007520 or A007521) of D. Then x^2 == 8 (mod p). Because the Legendre symbol (8/p) = (2*2^2/p) = (2/p) ==  (-1)^(p^2-1)/8 (see, e.g., Nagell, eq. (3), p. 138) this becomes -1 for these primes p, and therefore a candidate for D cannot have any prime factors 3 or 5 (mod 8).
However, not all of these candidates admit solutions. For the exceptions see A264348.
The remaining Ds (that admit solutions) are given in A263012.

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A263012, A264348.

Select[8 Range@ 154 + 1, Or[PrimeQ@ #, CompositeQ@ # && AllTrue[Union@ Mod[First /@ FactorInteger@ #, 8], ! MemberQ[{3, 5}, #] &]] && ! IntegerQ@ Sqrt@ # &] (* Michael De Vlieger, Dec 11 2015, Version 10 *)

A275793 The x members of the positive proper solutions (x = x1(n), y = y1(n)) of the first class for the Pell equation x^2 - 2*y^2 = +7^2.

Original entry on oeis.org

9, 43, 249, 1451, 8457, 49291, 287289, 1674443, 9759369, 56881771, 331531257, 1932305771, 11262303369, 65641514443, 382586783289, 2229879185291, 12996688328457, 75750250785451, 441504816384249, 2573278647520043, 14998167068736009, 87415723764896011

Offset: 0

Wolfdieter Lang, Sep 27 2016

This gives the (increasingly sorted) positive x members of the first class of the proper solutions (x1(n), y1(n)) to the Pell equation x^2 - 2*y^2 = +7^2. For the y1(n) solutions see 2*A275794(n). The solutions for the second class (x2(n), y2(n)) are found in A275795(n) and 2*A275796(n).
All solutions, including the improper ones, are given in A106525(n) and 2*A276600(n+2).
See also the comments on A263012 which apply here mutatis mutandis.
This is for the Pell equations x^2 - 2*y^2 = z^2, besides z^2 = 1 the first instance with proper solutions. For z^2 > 1 there seem to be always two classes of such solutions. For z^2 = 1 there is only one class of proper solutions. These z^2 values seem to appear for z from A058529 (prime factors are +1 or -1 (mod 8)).

			The first positive proper fundamental solution (x = x1(n), y = y1(n)) of x^2 - 2*y^2  = 49 are [9, 4], [43, 30], [249, 176], [1451, 1026], [8457, 5980], [49291, 34854], [287289, 203144], [1674443, 1184010], ...
The first positive proper fundamental solution of the second class (x = x2(n), y = y2(n)) are [11, 6], [57, 40], [331, 234], [1929, 1364], [11243, 7950], [65529, 46336], [381931, 270066], [2226057, 1574060], ...

T. Nagell, Introduction to Number Theory, Wiley, 1951, Theorem 109, pp. 207-208.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A000129, A001109, A058529, A106525.

Cf. A263012, A275794, A275795, A275796, 2*A276600.

I:=[9,43]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..31]]; // G. C. Greubel, Sep 15 2021

RecurrenceTable[{a[n]== 6a[n-1] -a[n-2], a[-1]==11, a[0]==9}, a, {n,0,25}] (* Michael De Vlieger, Sep 28 2016 *)
Table[9*Fibonacci[2*n+1, 2] - Fibonacci[2*n, 2], {n,0,30}] (* G. C. Greubel, Sep 15 2021 *)

a(n) = round((((3-2*sqrt(2))^n*(-8+9*sqrt(2))+(3+2*sqrt(2))^n*(8+9*sqrt(2)))) / (2*sqrt(2))) \\ Colin Barker, Sep 28 2016

Vec((9-11*x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Oct 02 2016

def P(n): return lucas_number1(n, 2, -1);
[9*P(2*n+1) - P(2*n) for n in (0..30)] # G. C. Greubel, Sep 15 2021

a(n) = 43*S(n-1, 6) - 9*S(n-2, 6), with the Chebyshev polynomials S(n, 6) = A001109(n+1), n >= -1, with S(-2, 6) = -1.
O.g.f: (9 - 11*x)/(1 - 6*x + x^2).
a(n) = 6*a(n-1) - a(n-2) for n >= 1, with a(-1) = 11 and a(0) = 9.
a(n) = (((3-2*sqrt(2))^n*(-8+9*sqrt(2))+(3+2*sqrt(2))^n*(8+9*sqrt(2)))) / (2*sqrt(2)). - Colin Barker, Sep 28 2016
a(n) = 9*A000129(2*n+1) - A000129(2*n). - G. C. Greubel, Sep 15 2021

A264354 Even nonsquare D values which admit proper solutions to the Pell equation x^2 - D*y^2 = +8.

Original entry on oeis.org

8, 28, 56, 92, 124, 136, 184, 188, 248, 284, 316, 376, 392, 412, 476, 508, 568, 604, 632, 668, 764, 776, 796, 824, 892, 952, 956, 1016, 1052, 1084, 1148, 1208, 1244, 1288, 1336, 1372, 1436, 1468, 1528, 1532

Offset: 1

Wolfdieter Lang, Nov 18 2015

nonn

This is 4*A261246.
The proper positive fundamental solution (x1(n), y1(n)) of the first class is given by (2*A261247(n), A261248(n)) for D(n) = a(n), n >= 1. If there are two classes the proper positive fundamental solution (x2(n), y2(n)) for the second class is given by (A264357(n), A264386(n)) for D(n). If the fundamental solutions of the two classes coincide then there is only one class (the ambiguous case) for these D(n) values. It is conjectured that there are no more than two classes. For the computation of (x2(n), y2(n)) from (x1(n), -y1(n)) by application of the matrix M(n) for D(n) see a comment under A263012.
D = 8, 56, 136, 184, 248, 376, 392, 568, 632, 776, 824, 952, 1016, 1208, 1288, 1336, 1528, ... have only one class of solution, because for them (x1, y1) = (x2, y2). These D values are the ones with x1(n) = 2*sqrt(x0(n)+1) and y1(n) = 2*y0(n) / sqrt(x0(n)+1) where (x0(n), y0(n)) are the positive fundamental solution of the +1 Pell equation with D = D(n). These are the upper bounds of the inequalities, eqs. (4) and (5) given in the Nagell reference on p. 206. E.g., D = 184 = A000037(171) = a(8) with x0(8) = A033313(171) = 24335 and y0(8) = A033317(171) = 1794 leads to x1(8) = 2*sqrt(24336) = 312 and y1(8) = 2*1794/sqrt(24336) = 23. These D numbers with only one class of proper solutions are the entries which are divisible by 8, that is four times the even numbers of A261246.

			The first positive proper fundamental solutions of the first class are, when written as [D(n), (x1(n), y1(n))]:
[8, (4, 1)], [28, (6, 1)], [56, (8, 1)], [92, (10, 1)], [124, (78, 7)], [136, (12, 1)], [184, (312, 23)], ...
The first positive proper fundamental solutions of the second class [D(n), (x2(n), y2(n))] are (if the values for both classes coincide there is only one class):
[8, (4, 1)], [28, (90, 17)], [56, (8, 1)], [92, (470, 49)], [124, (237042, 21287)], [136, (12, 1)], [184, (312, 23)], ...

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New Tork, 1964, p. 206.

Cf. A000037, A033313, A033317, A261246, A263012 (odd D), A261247 (x1/2), A261248 (y1), A264438 (x2), A264439 (y2), A264355.

a(n) = 4*A261246(n).

A264348 Exceptional odd numbers D that do not admit a solution to the Pell equation x^2 - D y^2 = +8 with both x and y odd.

Original entry on oeis.org

257, 401, 577, 697, 761, 1009, 1129, 1297, 1393, 1489, 1601, 1897, 2081, 2153, 2177, 2329, 2713, 2777, 2857, 2993, 3121, 3137, 3281, 3889, 4001

Offset: 1

Wolfdieter Lang, Nov 17 2015

nonn

These are the odd numbers 1 (mod 8), not a square, having in the composite case no prime factors 3 or 5 (mod 8), and the indefinite binary quadratic form x^2 - D*y^2 (with discriminant 4*D > 0) does not represent +8.

The odd numbers D which admit proper solutions to the Pell equation x^2 - D*y^2 = +8 with both x and y odd are given by A263012.

Cf. A263011, A263012.

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

A264351 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 odd D(n) = A263012(n).

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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).

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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).

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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).

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A263011 Numbers D == 1 (mod 8), not a square, and if composite without prime factors 3 or 5 (mod 8).

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A275793 The x members of the positive proper solutions (x = x1(n), y = y1(n)) of the first class for the Pell equation x^2 - 2*y^2 = +7^2.

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A264354 Even nonsquare D values which admit proper solutions to the Pell equation x^2 - D*y^2 = +8.

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A264348 Exceptional odd numbers D that do not admit a solution to the Pell equation x^2 - D y^2 = +8 with both x and y odd.

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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).

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