A255246 -id:A255246 - cp's OEIS frontend

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).

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

A254935 Fundamental positive solution y = y1(n) of the first 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

3, 5, 7, 7, 7, 9, 9, 11, 11, 11, 13, 15, 13, 13, 17, 15, 17, 19, 15, 17, 21, 17, 17, 21, 19, 23, 19, 19, 21, 23, 25, 21, 21, 27, 23, 29, 23, 23, 23, 23, 27, 25, 29, 31, 25, 33, 25, 27, 31, 29, 35, 27, 27, 31, 35, 33, 29, 35, 29, 31, 35, 31, 37, 31, 31, 33, 31, 41, 43, 39, 35, 37, 33, 41, 33, 35, 41

Offset: 1

Wolfdieter Lang, Feb 18 2015

For the corresponding term x1(n) see A254934(n).
See A254934 also for the Nagell reference.
The least positive y solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including prime 2) are given in A255246.

			See A254934.
n = 3: 5^2 - 2*7^2 = 25 - 98 = -73.

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. A254934 (corresponding x1 values), A254936 (x2 values), A254937 (y2 values), A254938 (same for primes == 7 mod 8), A255232 (y2 values, halved).

apply( {A254935(n, p=A007519(n))=sqrtint((A254934(,p)^2+p)\2)}, [1..77]) \\ M. F. Hasler, May 22 2025

A254934(n)^2 - 2*a(n)^2 = -A007519(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.

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

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

A255248 Fundamental positive solution y = y2(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

4, 7, 6, 10, 9, 8, 16, 11, 10, 15, 19, 14, 13, 22, 17, 16, 14, 24, 19, 28, 16, 27, 22, 31, 21, 26, 20, 19, 24, 29, 37, 21, 20, 32, 36, 31, 25, 30, 23, 22, 43, 29, 34, 28, 38, 42, 25, 45, 49, 29, 40, 35, 28, 27, 34, 39, 52, 43, 42, 28, 36, 46, 41, 35, 33, 32

Offset: 1

Wolfdieter Lang, Feb 19 2015

For the corresponding term x2(n) see A255247(n).

See the comments on A255247.

			See A255247.
a(4) = -(2*1 - 3*4) = 12 - 2 = 10.
n=4: 13^2 - 2*10^2 = 169 - 200 = -31 = -A001132(4).

Cf. A001132, A255247, A255235, A255246, A254937, A255234, A254931.

A255247(n)^2 - 2*a(n)^2 = -A001132(n), n >= 1, gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(2*A255235(n+1) - 3*A255246(n+1)), n >= 1.

More terms from Colin Barker, Feb 26 2015

cp's OEIS Frontend

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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A254935 Fundamental positive solution y = y1(n) of the first 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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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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A255248 Fundamental positive solution y = y2(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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