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

1,1

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.

Links

Table of n, a(n) for n=1..49.

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.

Example

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.

Sequence in context: A266731 A104995 A107744 * A275681 A173928 A304995

Adjacent sequences: A254759 A254760 A254761 * A254763 A254764 A254765

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 10 2015

Status

approved