A254937 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).

7, 9, 11, 15, 19, 13, 17, 19, 27, 31, 21, 19, 29, 37, 21, 31, 25, 23, 43, 29, 25, 45, 49, 29, 35, 27, 39, 43, 41, 35, 33, 53, 61, 35, 47, 33, 51, 55, 59, 63, 43, 53, 41, 39, 61, 37, 73, 55, 43, 49, 39, 67, 71, 51, 43, 49, 69, 47, 77, 63, 51, 67, 49, 71, 79, 65, 87, 49, 47, 55, 67, 61, 85, 53

Offset: 1

Author

Wolfdieter Lang, Feb 18 2015

Keywords

• nonn
• easy

Comments

The corresponding positive fundamental solution x2(n) of this second class solutions is given in A254936(n).

See the comments and the Nagell reference in A254934.

Examples

			n = 2: 11^2 - 2*9^2 = 121 - 162 = -41.
a(2) = -(2*3 - 3*5) = 9.
See also A254936.
		

Links

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

Crossrefs

Cf. A007519 (primes == 1 mod 8), A254936 (x2-values), A254934 (first (class) solution x1), A254935 (y1), A255234 (y2/2 for primes == 7 mod 8), A255248, A254760.

Programs

PARI

apply( {A254937(n, p=A007519(n))=n=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-2, 3]~}, [1..77]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

Formula

A254936(n)^2 - 2*a(n)^2 = -A007519(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(2*A254934(n) - 3*A254935(n)), n >= 1.

Extensions

More terms from M. F. Hasler, May 22 2025