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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.
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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)], ...
Positive integers D such that the generalized Pell equation X^2 - D Y^2 = 2 is soluble.
+10
8
2, 7, 14, 23, 31, 34, 46, 47, 62, 71, 79, 94, 98, 103, 119, 127, 142, 151, 158, 167, 191, 194, 199, 206, 223, 238, 239, 254, 263, 271, 287, 302, 311, 322, 334, 343, 359, 367, 382, 383, 386, 391, 398, 431, 439, 446, 463, 478, 479, 482, 487, 503, 511
COMMENTS
For the fundamental positive solution x(n)^2 - a(n)*y(n)^2 = 2 see (x(n) = A261247(n), y(n) = A261248(n)), for n >= 1. Conjecture: The sequence consists of all numbers D not a square and even D = 2*d has odd d with prime factors of the form 1 or 7 (mod 8). Odd D has prime factors of the form 1 or 7 (mod 8) but there is an odd number of primes of the form 7 (mod 8). The following will prove that these conditions for D are necessary in order to have solutions.
If there is a solution for D, D not a square, then only one class of solution exists due to Nagell's Theorem 110, p. 208, because then 2 divides 2*D. All solutions will be proper because 2 is a prime.
For the even prime D = p = 2 the positive fundamental solution is [x(1) = 2, y(1) = 1].
For odd primes D = p there can be solutions only for p == +7 (mod 8), that is p from A007522. Then x and y are both odd. Proof: Consider a solution of x^2 - p*y^2 = 2. The parities of x and y have to be either even and even or odd and odd. For odd x one has x^2 == +1 (mod 8) (because x^2 = 8*T(X) + 1 with x = 2*X+1 and the triangular numbers T = A000217); similarly for y^2 if y is odd. In the even-even case x^2 and y^2 are both congruent to 4 (mod 8). The even-even case leads to 4 - 4*p = 2 (mod 8), excluding all odd p, namely p == 1, 3, 5, 7 (mod 8). The odd-odd case is 1 - p*1 = 2 (mod 8), and p == 1, 3, 5 (mod 8) are excluded. Therefore, only p == 7 (mod 8) qualifies for a solution, and then x and y will be both odd. For D = p == 7 (mod 8) from A007522 one can test if there exists a fundamental positive solution (at most one class can exist, therefore there is either no solution or just one) [2*U(p)+1, 2*V(p)+1] by checking the two inequalities (see Nagell, eq. (4) and (5), p. 206) 0 <= V(p) < floor((Y(p)/sqrt(X(p) + 1) - 1)/2) and 0 <= U(p) <= floor((sqrt(X(p) + 1) - 1)/2), with the positive fundamental solution [X(p), Y(p)] of X^2 - p*Y^2 = +1. These solutions can be found in ( A033313(k), A033317(k)) if A000037(k) is the prime p == 7 (mod 8) one is testing. For composite even D there are solutions only if D/2 is odd. Proof: If D is even then x has to be even, hence x^2 == 0 (mod 4) and then D*y^2 == -2 (mod 4), hence D cannot be 0 (mod 4). Thus an even D can only be of the form D = 2*d with d odd. The modulo 3 and modulo 5 argument used in the next case will show that d can have only prime factors of the form +1 or -1 (mod 8).
For composite odd D one finds like above that the even-even x and y case is excluded, and the odd-odd case needs D == -1 (mod 8) == 7 (mod 8). Hence a candidate for D is from A004771 - A007522. D cannot have any prime factor p of the form 3 or 5 (mod 8) because otherwise x^2 == 2 (mod p), but the Legendre symbol (2/p) = -1 for such p's (see, e.g., Nagell, Theorem 81, p. 136). For example, D = 15 = 3*5 cannot have a solution. Thus the only candidates for D have prime factors p of the form +1 or +7 (mod 8), with the number of the latter ones being odd. E.g., D = 7*17 = 119 qualifies as a candidate and it has indeed solutions, namely the ones obtainable from the fundamental one [11, 1]. The general proper positive solutions for D(n) = a(n) are obtained from the fundamental ones [x(n), y(n)] given in A261247 and A261248 with the help of powers of the matrix M(n) = [[u(n), D(n)*v(n)], [v(n), u(n)]], where u(n) and v(n) are the positive fundamental solutions of U(n) - D(n)*V(n) = 1, by (x(n; k), y(n; k))^T = M(n)^k (x(n), y(n))^T (T for transposed), for k >= 0. [u(n), v(n)] = [ A033313(j(n)), A033317(j(n))] if A000037(j(n)) = D(n) = a(n). Observation: All degrees (7, 47, 79, 103, 119, 127) of the modular equations derived for solving Ramanujan's question 699 by Galkin & Kozirev (see reference and A318732) are terms of this sequence. - Hugo Pfoertner, Sep 24 2023
REFERENCES
J. W. S. Cassels, Rational Quadratic Forms, Cambridge, 1978; see Chap. 3.
V. M. Galkin, O. R. Kozyrev, On an algebraic problem of Ramanujan, pp. 89-94 in Number Theoretic And Algebraic Methods In Computer Science - Proceedings Of The International Conference, Moscow 1993, Ed. Horst G. Zimmer, World Scientific, 31 Aug 1995
T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.
EXAMPLE
The first fundamental solutions [x(n), y(n)] are (the first entry gives D(n)=a(n)):
[2, [2, 1]], [7, [3, 1]], [14, [4, 1]],
[23, [5, 1]], [31, [39, 7]], [34, [6, 1]],
[46, [156, 23]], [47, [7, 1]], [62, [8, 1]],
[71, [59, 7]], [79, [9, 1]], [94, [1464, 151]],
[98, [10, 1]], [103, [477, 47]], [119, [11, 1]],
[127, [2175, 193]], [142, [12, 1]],
[151, [41571, 3383]], [158, [88, 7]],
[167, [13, 1]], [191, [2999, 217]],
[194, [14, 1]], [199, [127539, 9041]],
[206, [244, 17]], [223, [15, 1]], [238, [108, 7]],
[239, [2489, 161]], ...
MATHEMATICA
Select[Range[600], False =!= Reduce[x^2 - # y^2 == 2, {x, y}, Integers] &] (* Giovanni Resta, Aug 12 2017 *)
Exceptional odd numbers D that do not admit a solution to the Pell equation X^2 - D Y^2 = +2.
+10
2
791, 799, 943, 1271, 1351, 1631, 1751, 1967, 2159, 2303, 2359, 2567, 3143, 3199, 3503, 3703, 3983, 4063, 4439, 4471, 4559, 4607, 4711, 5047, 5183, 5207, 5359, 5663, 5911, 5983, 6511, 6671, 6839, 7063, 7231, 7663, 7871, 8183, 8407, 8711, 9143, 9271, 9751, 9863, 10183, 10367
COMMENTS
These are the odd numbers 7 (mod 8), not a square, that have in the composite case no prime factors 3 or 5 (mod 8), and do not represent +2 by the indefinite binary quadratic form X^2 - D*Y^2 (with discriminant 4*D > 0).
The numbers D which admit solutions of the Pell equation X^2 - D Y^2 = +2 are given by A261246. Necessary conditions for nonsquare odd D were shown there to be D == 7 (mod 8), without prime factors 3 or 5 (mod 8) in the composite case. Thus only prime factors +1 (mod 8) and -1 (mod 8) can appear, and the number of the latter is odd. It has been conjectured that all such numbers D appear in A261246, but this conjecture is false as the present sequence shows. All entries seem to be composite. The first numbers are 791 = 7*113, 799 = 17*47, 943 = 23*41, 1271 = 31*41, 1351 = 7*193, 1631 = 7*233, ...
For counterexamples to the conjecture in A261246 for even D see A264352.
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