A255248 -id:A255248 - cp's OEIS frontend

A254937 Fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007519(n), n>=1 (primes congruent to 1 mod 8).

7, 9, 11, 15, 19, 13, 17, 19, 27, 31, 21, 19, 29, 37, 21, 31, 25, 23, 43, 29, 25, 45, 49, 29, 35, 27, 39, 43, 41, 35, 33, 53, 61, 35, 47, 33, 51, 55, 59, 63, 43, 53, 41, 39, 61, 37, 73, 55, 43, 49, 39, 67, 71, 51, 43, 49, 69, 47, 77, 63, 51, 67, 49, 71, 79, 65, 87, 49, 47, 55, 67, 61, 85, 53

Offset: 1

Wolfdieter Lang, Feb 18 2015

The corresponding positive fundamental solution x2(n) of this second class solutions is given in A254936(n).

See the comments and the Nagell reference in A254934.

			n = 2: 11^2 - 2*9^2 = 121 - 162 = -41.
a(2) = -(2*3 - 3*5) = 9.
See also A254936.

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

Cf. A007519 (primes == 1 mod 8), A254936 (x2-values), A254934 (first (class) solution x1), A254935 (y1), A255234 (y2/2 for primes == 7 mod 8), A255248, A254760.

apply( {A254937(n, p=A007519(n))=n=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-2, 3]~}, [1..77]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

A254936(n)^2 - 2*a(n)^2 = -A007519(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(2*A254934(n) - 3*A254935(n)), n >= 1.

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

A255235 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A038873(n), n>=1 (primes congruent to {1,2,7} mod 8).

Original entry on oeis.org

4, 1, 1, 3, 1, 3, 5, 1, 5, 7, 3, 1, 5, 7, 1, 5, 7, 11, 3, 7, 1, 13, 3, 7, 1, 9, 5, 11, 13, 9, 5, 1, 15, 17, 5, 3, 7, 13, 9, 17, 19, 1, 11, 7, 13, 5, 3, 19, 3, 1, 17, 7, 11, 19, 21, 13, 9, 1, 7, 9, 25, 15, 7, 11, 17, 21, 23, 27, 5

Offset: 1

Wolfdieter Lang, Feb 25 2015

For the corresponding term y1(n) see A255246(n).
The present solutions of this first class are the smallest positive ones.
For the positive fundamental proper (sometimes called primitive) solutions x2 and y2 of the second class of this (generalized) Pell equation see A255247 and A255248. There is no second class for prime 2.
For the first class solutions of this Pell equation with primes 1 (mod 8) see A254934 and A254935. For those with primes 7 (mod 8) see A254938 and 2*A255232. For the derivation of these solutions see A254934 and A254938, also for the Nagell reference.

			The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are
  (the prime A038873(n) is listed as first entry):
  [2,[4, 3]], [7, [1, 2]], [17, [1, 3]],
  [23, [3, 4]], [31, [1, 4]], [41, [3, 5]],
  [47, [5, 6]], [71, [1, 6]], [73, [5, 7]],
  [79, [7, 8]], [89, [3, 7]], [97, [1, 7]],
  [103, [5, 8]], [113, [7, 9]], [127, [1, 8]],
  [137, [5, 9]], [151, [7, 10]], [167, [11, 12]], [191, [3, 10]], [193, [7, 11]], [199, [1, 10]], [223, [13, 14]], [233, [3, 11]], [239, [7, 12]], [241, [1, 11]], [257, [9, 13]], [263, [5, 12]], ...
n=1: 4^2 - 2*3^2 = -2 = -A038873(1),
n=2: 1^2 - 2*2^2 = 1 - 8 = -7 = -A038873(2).

Cf. A038873, A255246, A255247, A255248, A254934, A254935, A254938, 2*A255232, A002334.

a(n)^2 - A255246(n)^2 = - A038873(n), n >= 1, gives the smallest positive (proper) solution of this (generalized) Pell equation.

More terms from Colin Barker, Feb 26 2015

A255246 Fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A038873(n), n>=1 (primes congruent to {1,2,7} mod 8).

Original entry on oeis.org

3, 2, 3, 4, 4, 5, 6, 6, 7, 8, 7, 7, 8, 9, 8, 9, 10, 12, 10, 11, 10, 14, 11, 12, 11, 13, 12, 14, 15, 14, 13, 13, 17, 18, 14, 14, 15, 17, 16, 19, 20, 15, 17, 16, 18, 16, 16, 21, 17, 17, 21, 18, 19, 22, 23, 20, 19, 18, 19, 20, 26, 22, 20, 21, 23, 25, 26, 28, 21

Offset: 1

Wolfdieter Lang, Feb 25 2015

For the corresponding term x1(n) see A255235(n).
For the primes 1 (mod 8) see A154935, and for the primes 7 (mod 8) see 2*A255232.
See A254934 and A254938 also for the derivation based on the Nagell reference given there.

			See A255235.
n = 1: 4^2 - 2*3^2 = -2 = -A038873(1),
n = 3: 1^2 - 2*3^2 = 1 - 18 = -17 = -A038873(3).

Cf. A038873, A255235, A255247, A255248, A254935, 2*A255232, A002335.

A255235(n)^2 - 2*a(n)^2 = -A038873(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.

More terms from Colin Barker, Feb 26 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

A254937 Fundamental positive solution y = y2(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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A255235 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A038873(n), n>=1 (primes congruent to {1,2,7} mod 8).

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A255246 Fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A038873(n), n>=1 (primes congruent to {1,2,7} mod 8).

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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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