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, 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

Author

Wolfdieter Lang, Feb 19 2015

Keywords

• nonn

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.

Examples

			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.

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.

Extensions

More terms from Colin Barker, Feb 26 2015