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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). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..66.

FORMULA

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.

EXAMPLE

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.

CROSSREFS

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

Sequence in context: A232190 A160050 A055566 * A153610 A249385 A247747

Adjacent sequences:  A255244 A255245 A255246 * A255248 A255249 A255250

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Feb 19 2015

EXTENSIONS

More terms from Colin Barker, Feb 26 2015

STATUS

approved

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Last modified October 21 20:44 EDT 2019. Contains 328315 sequences. (Running on oeis4.)