A255233 - OEIS

A255233

Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

5, 7, 13, 9, 21, 11, 17, 29, 19, 15, 31, 37, 17, 27, 33, 23, 29, 21, 41, 47, 37, 23, 43, 33, 49, 55, 51, 31, 41, 69, 53, 29, 43, 59, 35, 31, 45, 61, 41, 67, 85, 57, 47, 63, 43, 53, 35, 75, 93, 37, 71, 61, 83, 47, 89, 39, 73, 53, 63, 79, 49, 85, 69, 97, 103, 109, 55, 65, 47, 77, 67, 83, 49

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OFFSET

1,1

COMMENTS

The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = -A007522(n) = -(1 + A139487(n)*8) is given in 2*A255234(n).

For comments and the Nagell reference see A254938.

LINKS

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

FORMULA

a(n)^2 - 2*(2*A255234(n))^2 = -A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(3*A254938(n) - 4*2*A255232(n)), n >= 1.

EXAMPLE

The first pairs [x2(n), y2(n)] of the fundamental positive solutions of this second class are (the prime A007522(n) appears as first entry):

[7, [5, 4]], [23, [7, 6]], [31, [13, 10]],

[47, [9, 8]], [71, [21, 16]], [79, [11, 10]], [103, [17, 14]], [127, [29, 22]],

[151, [19, 16]], [167, [15, 14]],

[191, [31, 24]], [199, [37, 28]],

[223, [17, 16]], [239, [27, 22]],

[263, [33, 26]], [271, [23, 20]],

[311, [29, 24]], [359, [21, 20]],

[367, [41, 32]], [383, [47, 36]],

[431, [37, 30]], [439, [23, 22]],

[463, [43, 34]], [479, [33, 28]], ...

n= 4: 9^2 - 2*(2*4)^2 = -47 = -A007522(4).

a(4) = -(3*5 - 4*(2*3)) = 24 - 15 = 9.

PROG

(PARI) apply( {A255233(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-3, 4]~}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

CROSSREFS

Cf. A007522 (primes == 7 mod 8), A139487 (8k+7 is prime).

Cf. 2*A255234 (corresponding y2 values), A254938 (x1 values), 2*A255232 (y2 values), A255247, A254936.

Sequence in context: A205323 A105054 A088897 * A361708 A074755 A354200

Adjacent sequences: A255230 A255231 A255232 * A255234 A255235 A255236

KEYWORD

nonn,look,easy

AUTHOR

Wolfdieter Lang, Feb 18 2015

EXTENSIONS

More terms from Colin Barker, Feb 23 2015

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

STATUS

approved