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

Offset: 1

Author

Wolfdieter Lang, Feb 18 2015

Keywords

• nonn
• look
• easy

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.

Examples

			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.
		

Links

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

Programs

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

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.

Extensions

More terms from Colin Barker, Feb 23 2015

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