Voozh

A254935

Fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = -A007519(n), n>=1 (primes congruent to 1 mod 8).

3, 5, 7, 7, 7, 9, 9, 11, 11, 11, 13, 15, 13, 13, 17, 15, 17, 19, 15, 17, 21, 17, 17, 21, 19, 23, 19, 19, 21, 23, 25, 21, 21, 27, 23, 29, 23, 23, 23, 23, 27, 25, 29, 31, 25, 33, 25, 27, 31, 29, 35, 27, 27, 31, 35, 33, 29, 35, 29, 31, 35, 31, 37, 31, 31, 33, 31, 41, 43, 39, 35, 37, 33, 41, 33, 35, 41

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For the corresponding term x1(n) see A254934(n).

See A254934 also for the Nagell reference.

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

LINKS

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

FORMULA

A254934(n)^2 - 2*a(n)^2 = -A007519(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.

EXAMPLE

See A254934.

n = 3: 5^2 - 2*7^2 = 25 - 98 = -73.

PROG

(PARI) apply( {A254935(n, p=A007519(n))=sqrtint((A254934(, p)^2+p)\2)}, [1..77]) \\ M. F. Hasler, May 22 2025

CROSSREFS

Cf. A007519 (primes == 1 mod 8), A005123 (8k+1 is prime).

Cf. A254934 (corresponding x1 values), A254936 (x2 values), A254937 (y2 values), A254938 (same for primes == 7 mod 8), A255232 (y2 values, halved).