A255232 One half of the fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

1, 2, 2, 3, 3, 4, 4, 4, 5, 6, 5, 5, 7, 6, 6, 7, 7, 9, 7, 7, 8, 10, 8, 9, 8, 8, 9, 11, 10, 9, 10, 13, 11, 10, 13, 14, 12, 11, 13, 11, 11, 12, 13, 12, 14, 13, 16, 12, 12, 17, 13, 14, 13, 16, 13, 18, 14, 16, 15, 14, 17, 14, 15, 14, 14, 14, 17, 16, 19, 16, 17, 16, 20, 21, 17, 16, 17, 16, 16

Offset: 1

Author

Wolfdieter Lang, Feb 18 2015

Keywords

• nonn
• easy

Comments

For the corresponding term x1(n) see A254938(n).

See A254938 also for the Nagell reference.

The least positive y solutions (that is those of the first class) for the primes +1 and -1 (mod 8) together (including prime 2) are given in A255246.

Examples

			See A254938.
n = 3: 1^2 - 2*(2*2)^2 = 1 - 32  = -31 = -A007522(3).
		

Links

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

Crossrefs

Cf. A007522 (primes == 7 mod 8), A254938 (corresponding x1 values), A255233 (x2 values), A255234 (y2 values), A255246, A254935.

Programs

PARI

apply( {A255232(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1][2]\2}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

Formula

A254938(n)^2 - 2*(2*a(n))^2 = -A007522(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.

Extensions

More terms from Colin Barker, Feb 23 2015

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