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).

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

Author

Wolfdieter Lang, Feb 25 2015

Keywords

• nonn
• easy

Comments

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.

Examples

			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).
		

Crossrefs

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

Formula

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

Extensions

More terms from Colin Barker, Feb 26 2015