A264348 -id:A264348 - 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.

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 *)

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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.

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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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