A255233 -id:A255233 - cp's OEIS frontend

A254938 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

1, 3, 1, 5, 1, 7, 5, 1, 7, 11, 3, 1, 13, 7, 5, 11, 9, 17, 5, 3, 9, 19, 7, 13, 5, 3, 7, 19, 13, 1, 9, 25, 15, 7, 23, 27, 17, 9, 21, 7, 1, 13, 19, 11, 23, 17, 31, 7, 1, 33, 11, 17, 7, 27, 5, 35, 13, 25, 19, 11, 29, 9, 17, 5, 3, 1, 27, 21, 35, 17, 23, 15, 37, 41, 21, 13, 19, 7, 3, 23, 15, 33, 13

Offset: 1

Wolfdieter Lang, Feb 18 2015

nonn

For the corresponding term y1(n) see 2*A255232(n).
For the positive fundamental proper (sometimes called primitive) solutions x2(n) and y2(n) of the second class of this (generalized) Pell equation see A255233(n) and A255234(n).
The present solutions of this first class are the smallest positive ones.
See the Nagell reference Theorem 111, p. 210, for the proof of the existence of solutions (the discriminant of this binary quadratic form is +8 hence it is an indefinite form with an infinitude of solutions if there exists at least one).
See the Nagell reference Theorem 110, p. 208, for the proof that there are only two classes of solutions for this Pell equation, because the equation is solvable and each prime from A007522 does not divide 4.
The present fundamental solutions are found according to the Nagell reference Theorem 108a, p. 206-207, adapted to the case at hand, by scanning the following two inequalities for solutions x1(n) = 2*X1(n) + 1 and y1(n) = 2*Y1(n) (because odd y is out in this Pell equation). The interval to be scanned for X1(n) is [0, floor((sqrt(p(n))-1)/2)] and for Y1(n) it is [0, floor(sqrt(p(n))/2)], with p(n) = A007522(n).
The general positive proper solutions are for both classes obtained by applying positive powers of the matrix M = [[3,4],[2,3]] on the fundamental positive column vectors (x(n),y(n))^T. The n-th power is M^n = S(n-1, 6)*M - S(n-2, 6) 1_2, where 1_2 is the 2 X 2 identity matrix and S(n, 6), with S(-2, 6) = -1 and S(-1, 6) = 0, is the Chebyshev S-polynomial evaluated at x = 6, given in A001109(n).
The least positive x solutions (that is those of the first class) for the primes +1 and -1 (mod 8) together (including prime 2) are given in A255235.

			The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (the prime A007522(n) is listed as first entry):
  [7, [1, 2]], [23, [3, 4]], [31, [1, 4]],
  [47, [5, 6]], [71, [1, 6]], [79, [7, 8]],
  [103, [5, 8]], [127, [1, 8]], [151, [7, 10]],
  [167, [11, 12]], [191, [3, 10]], [199, [1, 10]], [223, [13, 14]], [239, [7, 12]], [263, [5, 12]], [271, [11, 14]], [311, [9, 14]], [359, [17, 18]], [367, [5, 14]], [383, [3, 14]], [431, [9, 16]], [439, [19, 20]], [463, [7, 16]], [479, [13, 18]], [487, [5, 16]], [503, [3, 16]], ...
n=1: 1^2 - 2*(2*1)^2 = 1 - 8 = -7 = -A007522(1), ...

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

M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

Cf. A007522 (primes == 7 mod 8).
Cf. A255232 (corresponding y2 values, halved), A255233 (x2 values), A255234 (y2 values), A255235.
Cf. A254934 - A254937 (similar for primes == 1 mod 8).

apply( {A254938(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1][1]}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

a(n)^2 - 2*(2*A255232(n))^2 = -A007522(n), n >= 1, gives the smallest positive (proper) solution of this (generalized) Pell equation.

More terms from Colin Barker, Feb 23 2015

A255232 One half of the fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 5, 6, 5, 5, 7, 6, 6, 7, 7, 9, 7, 7, 8, 10, 8, 9, 8, 8, 9, 11, 10, 9, 10, 13, 11, 10, 13, 14, 12, 11, 13, 11, 11, 12, 13, 12, 14, 13, 16, 12, 12, 17, 13, 14, 13, 16, 13, 18, 14, 16, 15, 14, 17, 14, 15, 14, 14, 14, 17, 16, 19, 16, 17, 16, 20, 21, 17, 16, 17, 16, 16

Offset: 1

Wolfdieter Lang, Feb 18 2015

For the corresponding term x1(n) see A254938(n).
See A254938 also for the Nagell reference.
The least positive y solutions (that is those of the first class) for the primes +1 and -1 (mod 8) together (including prime 2) are given in A255246.

			See A254938.
n = 3: 1^2 - 2*(2*2)^2 = 1 - 32  = -31 = -A007522(3).

M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

Cf. A007522 (primes == 7 mod 8), A254938 (corresponding x1 values), A255233 (x2 values), A255234 (y2 values), A255246, A254935.

apply( {A255232(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1][2]\2}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

A254938(n)^2 - 2*(2*a(n))^2 = -A007522(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.

More terms from Colin Barker, Feb 23 2015

Double-checked and extended by M. F. Hasler, May 22 2025

A254936 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007519(n), n >= 1 (primes congruent to 1 mod 8).

Original entry on oeis.org

9, 11, 13, 19, 25, 15, 21, 23, 35, 41, 25, 21, 37, 49, 23, 39, 29, 25, 57, 35, 27, 59, 65, 33, 43, 29, 49, 55, 51, 41, 37, 69, 81, 39, 59, 35, 65, 71, 77, 83, 51, 67, 47, 43, 79, 39, 97, 69, 49, 59, 41, 87, 93, 61, 47, 57, 89, 53, 101, 79, 59, 85, 55, 91, 103, 81, 115, 53, 49, 63, 83, 73, 111, 59

Offset: 1

Wolfdieter Lang, Feb 18 2015

The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = -A007519(n) = -(1 + 8*A005123(n)) is given in A254937(n).

For comments and the Nagell reference see A254934.

			The first pairs [x2(n), y2(n)] of the fundamental positive solutions of this second class are (the prime A007519(n) appears as first entry):
  [17, [9, 7]], [41, [11, 9]], [73, [13, 11]],
  [89, [19, 15]], [97, [25, 19]], [113, [15, 13]],
  [137, [21, 17]], [193, [23, 19]], [233, [35, 27]],
  [241, [41, 31]], [257, [25, 21]], [281, [21, 19]],
  [313, [37, 29]], [337, [49, 37]], [353, [23, 21]],
  [401, [39, 31]], [409, [29, 25]], [433, [25, 23]],
  [449, [57, 43]], [457, [35, 29]], [521, [27, 25]],
  [569, [59, 45]], [577, [65, 49]], [593, [33, 29]],
  [601, [43, 35]], [617, [29, 27]], [641, [49, 39]], ...
a(4) = -(3*3 - 4*7) = 28 - 9 = 19.

M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

Cf. A007519 (primes == 1 mod 8), A005123 (8k+1 is prime).

Cf. A254937 (corresponding y2-values), A254934 (x1 values), A254935 (y1 values), A255233 (same for primes == 7 mod 8), A255247.

apply( {A254936(n, p=A007519(n))=n=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-3,4]~}, [1..77]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

a(n)^2 - 2*A254937(n)^2 = -A007519(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(3*A254934(n) - 4*A254935(n)), n >= 1.

More terms from M. F. Hasler, May 22 2025

A255234 One half of the fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n>=1 (primes congruent to 7 mod 8).

Original entry on oeis.org

2, 3, 5, 4, 8, 5, 7, 11, 8, 7, 12, 14, 8, 11, 13, 10, 12, 10, 16, 18, 15, 11, 17, 14, 19, 21, 20, 14, 17, 26, 21, 14, 18, 23, 16, 15, 19, 24, 18, 26, 32, 23, 20, 25, 19, 22, 17, 29, 35, 18, 28, 25, 32, 21, 34, 19, 29, 23, 26, 31, 22, 33, 28, 37, 39, 41, 24, 27, 22, 31, 28, 33, 23, 22, 30

Offset: 1

Wolfdieter Lang, Feb 19 2015

The corresponding fundamental solution x2(n) of this second class of positive solutions is given in A255233(n).

See the comments and the Nagell reference in A254938.

			n = 2: 7^2 - 2*(2*3)^2 = 49 - 72  = -23 = - A007522(2).
a(3) = -(1 - 3*2) = 5.
See also A255233.

M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

Cf. A007522, A255233, A254938, A255232, A254937.

apply( {A255234(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-1,3/2]~}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

A255233(n)^2 - 2*(2*a(n))^2 = -A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(A254938(n) - 3*A255232(n)), n >= 1.

More terms from Colin Barker, Feb 24 2015

Double-checked and extended by M. F. Hasler, May 22 2025

A255236 All positive solutions x of the second class of the Pell equation x^2 - 2*y^2 = -7.

Original entry on oeis.org

5, 31, 181, 1055, 6149, 35839, 208885, 1217471, 7095941, 41358175, 241053109, 1404960479, 8188709765, 47727298111, 278175078901, 1621323175295, 9449763972869, 55077260661919, 321013799998645, 1871005539329951, 10905019435981061, 63559111076556415

Offset: 0

Wolfdieter Lang, Feb 26 2015

For the corresponding y = y2 terms see 2*A038725(n+1).
The Pell equation x^2 - 2*y^2 = 7 has two classes of solutions. See, e.g., the Nagell reference and comments under A254938 and A255233. Here the positive solutions based on the fundamental solution (5, 4) (the second largest positive solution) are considered.
The positive solutions of the first class are given in (A054490(n), 2*A038723(n)), n >= 0.
The combined solutions of both classes are given in (A077446, 4*A077447).
The solutions (x(n), y(n)) of x^2 - 2*y^2 = -7 translate to the solutions (X(n), Y(n)) = (2*y(n) , x(n)) of the Pell equation X^2 - 2*Y^2 = 14.

			n = 2: 181^2 - 2*(2*64)^2  = -7; (4*64)^2 - 2*181^2 = 14.
n = 2: 2*53 + 75 = 181. - _Wolfdieter Lang_, Mar 19 2015

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. A038725, A054490, A038723, A077446, 4*A077447, A254938, A255233.

I:=[5,31]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 20 2015

CoefficientList[Series[(5 + x) / (1 - 6 x + x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 20 2015 *)

Vec((5 + x)/(1 - 6*x + x^2) + O(x^30)) \\ Michel Marcus, Mar 20 2015

a(n) = 5*S(n, 6) + S(n-1, 6), n >= 0, with the Chebyshev polynomials S(n, x) (A049310), with S(-1, x) = 0, evaluated at x = 6. S(n, 6) = A001109(n-1).
G.f.: (5 + x)/(1 - 6*x + x^2).
a(n) = 6*a(n-1) - a(n-2), n >= 2, with a(-1) = -1 and a(0) = 5.
a(n) = 2*A038761(n) + A038762(n), n >= 0. See the Mar 19 comment on A054490. - Wolfdieter Lang, Mar 19 2015
a(n) = ((3-2*sqrt(2))^n*(-8+5*sqrt(2)) + (3+2*sqrt(2))^n*(8+5*sqrt(2))) / (2*sqrt(2)). - Colin Barker, Oct 13 2015

A255247 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A001132(n), n>=1 (primes congruent to {1,7} mod 8).

Original entry on oeis.org

5, 9, 7, 13, 11, 9, 21, 13, 11, 19, 25, 17, 15, 29, 21, 19, 15, 31, 23, 37, 17, 35, 27, 41, 25, 33, 23, 21, 29, 37, 49, 23, 21, 41, 47, 39, 29, 37, 25, 23, 57, 35, 43, 33, 49, 55, 27, 59, 65, 33, 51, 43, 31, 29, 41, 49, 69, 55, 53, 29, 43, 59, 51, 41, 37, 35

Offset: 1

Wolfdieter Lang, Feb 19 2015

nonn

For the corresponding term y2(n) see A255248(n).
For the positive fundamental proper (sometimes called primitive) solutions x1(n) and y1(n) of the first class of this (generalized) Pell equation see A255235(n) and A255246(n).
The present solutions of this second class are the next to smallest positive ones. Note that for prime 2 only the first class exists.
For the derivation based on the book of Nagell see the comments on A254934 and A254938 for the primes 1 (mod 8) and 7 (mod 8) separately, where also the Nagell reference is given.

			The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (the prime A001132(n) is listed as first entry):
  [7, [5, 4]], [17, [9, 7]], [23, [7, 6]],
  [31, [13, 10]], [41, [11, 9]], [47, [9, 8]],
  [71, [21, 16]], [73, [13, 11]], [79, [11, 10]],
  [89, [19, 15]], [97, [25, 19]], [103, [17, 14]],
  [113, [15, 13]], [127, [29, 22]],
  [137, [21, 17]], [151, [19, 16]],
  [167, [15, 14]], [191, [31, 24]],
  [193, [23, 19]], [199, [37, 28]],
  [223, [17, 16]], [233, [35, 27]],
  [239, [27, 22]], [241, [41, 31]], ...
n = 1: 5^2 - 2*4^2 = 25 - 32 = -7 = -A001132(1).
a(3) = -(3*3 - 4*4) = 16 - 9 = 7.

Cf. A001132, A255248, A255235, A255246, A254936, A255233, A254930.

a(n)^2 - 2*A255248(n)^2 = -A001132(n), n >= 1, gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(3*A255235(n+1) - 4*A255246(n+1)), n >= 1.

More terms from Colin Barker, Feb 26 2015

cp's OEIS Frontend

A254938 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

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A255232 One half of the fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

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A254936 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007519(n), n >= 1 (primes congruent to 1 mod 8).

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A255234 One half of the fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n>=1 (primes congruent to 7 mod 8).

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A255236 All positive solutions x of the second class of the Pell equation x^2 - 2*y^2 = -7.

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A255247 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A001132(n), n>=1 (primes congruent to {1,7} mod 8).

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