A254762 Fundamental positive solution x = x2(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, 13, 19, 17, 15, 25, 23, 29, 25, 23, 35, 43, 31, 27, 49, 37, 47, 55, 31, 45, 61, 37, 35, 59, 49, 67, 47, 45, 53, 63, 71, 47, 43, 77, 57, 85, 55, 53, 51, 49, 73, 61, 81, 89, 57, 97, 51, 67, 87

Offset: 1

Author

Wolfdieter Lang, Feb 10 2015

Keywords

• nonn
• easy

Comments

The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = A007519(n) = 1 + 8*A005123(n) is given in 2*A254763(n).

For comments and the Nagell reference see A254760.

Examples

			The first pairs [x2(n), y2(n)] of the fundamental positive solutions of the second class are (we list the prime A007519(n) as first entry):
  [17, [7, 4]], [41, [13, 8]], [73, [19, 12]], [89, [17, 10]], [97, [15, 8]], [113, [25, 16]], [137, [23, 14]], [193, [29, 18]], [233, [25, 14]], [241, [23, 12]], [257, [35, 22]], [281, [43, 28]], [313, [31, 18]], [337, [27, 14]], [353, [49, 32]], [401, [37, 22]], [409, [47, 30]], ...
a(4) = 3*11 - 8*2 = 17.
		

Crossrefs

Cf. A007519, A005123, 2*A254763, A254760, 2*A254761.

Formula

a(n)^2 - 2*(2*A254763(n))^2 = A007519(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = 3*A254760(n) - 8*A254761(n), n >= 1.