A255234 One half of the fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n>=1 (primes congruent to 7 mod 8).

2, 3, 5, 4, 8, 5, 7, 11, 8, 7, 12, 14, 8, 11, 13, 10, 12, 10, 16, 18, 15, 11, 17, 14, 19, 21, 20, 14, 17, 26, 21, 14, 18, 23, 16, 15, 19, 24, 18, 26, 32, 23, 20, 25, 19, 22, 17, 29, 35, 18, 28, 25, 32, 21, 34, 19, 29, 23, 26, 31, 22, 33, 28, 37, 39, 41, 24, 27, 22, 31, 28, 33, 23, 22, 30

Offset: 1

Author

Wolfdieter Lang, Feb 19 2015

Keywords

• nonn
• look
• easy

Comments

The corresponding fundamental solution x2(n) of this second class of positive solutions is given in A255233(n).

See the comments and the Nagell reference in A254938.

Examples

			n = 2: 7^2 - 2*(2*3)^2 = 49 - 72  = -23 = - A007522(2).
a(3) = -(1 - 3*2) = 5.
See also A255233.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

Crossrefs

Cf. A007522, A255233, A254938, A255232, A254937.

Programs

PARI

apply( {A255234(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-1,3/2]~}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

Formula

A255233(n)^2 - 2*(2*a(n))^2 = -A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(A254938(n) - 3*A255232(n)), n >= 1.

Extensions

More terms from Colin Barker, Feb 24 2015

Double-checked and extended by M. F. Hasler, May 22 2025