A254763 One half of the fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = A007519(n), n>=1 (primes congruent to 1 mod 8).

2, 4, 6, 5, 4, 8, 7, 9, 7, 6, 11, 14, 9, 7, 16, 11, 15, 18, 8, 14, 20, 10, 9, 19, 15, 22, 14, 13, 16, 20, 23, 13, 11, 25, 17, 28, 16, 15, 14, 13, 23, 18, 26, 29, 16, 32, 13, 20, 28, 24

Offset: 1

Author

Wolfdieter Lang, Feb 10 2015

Keywords

• nonn
• easy

Comments

The corresponding fundamental solution x2(n) of this second class of positive solutions is given in A254762(n).

See the comments and the Nagell reference in A254760.

Examples

			n = 2: 13^2 - 2*(2*4)^2 = 169 - 128 = 41.
The smallest positive solution is (x1(2), y1(2)) = (7, 2) from (A254760(2), 2*A254761(2)).
See also A254762.
a(4) = 11 - 3*2 = 5.
		

Crossrefs

Cf. A007519, A005123, A254762, A254760, 2*A254761.

Formula

A254762(n)^2 - 2*(2*a(n))^2 = A007519(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

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