A254936 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007519(n), n >= 1 (primes congruent to 1 mod 8).

9, 11, 13, 19, 25, 15, 21, 23, 35, 41, 25, 21, 37, 49, 23, 39, 29, 25, 57, 35, 27, 59, 65, 33, 43, 29, 49, 55, 51, 41, 37, 69, 81, 39, 59, 35, 65, 71, 77, 83, 51, 67, 47, 43, 79, 39, 97, 69, 49, 59, 41, 87, 93, 61, 47, 57, 89, 53, 101, 79, 59, 85, 55, 91, 103, 81, 115, 53, 49, 63, 83, 73, 111, 59

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 = -A007519(n) = -(1 + 8*A005123(n)) is given in A254937(n).

For comments and the Nagell reference see A254934.

Links

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

Formula

a(n)^2 - 2*A254937(n)^2 = -A007519(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(3*A254934(n) - 4*A254935(n)), n >= 1.

Example

The first pairs [x2(n), y2(n)] of the fundamental positive solutions of this second class are (the prime A007519(n) appears as first entry):
  [17, [9, 7]], [41, [11, 9]], [73, [13, 11]],
  [89, [19, 15]], [97, [25, 19]], [113, [15, 13]],
  [137, [21, 17]], [193, [23, 19]], [233, [35, 27]],
  [241, [41, 31]], [257, [25, 21]], [281, [21, 19]],
  [313, [37, 29]], [337, [49, 37]], [353, [23, 21]],
  [401, [39, 31]], [409, [29, 25]], [433, [25, 23]],
  [449, [57, 43]], [457, [35, 29]], [521, [27, 25]],
  [569, [59, 45]], [577, [65, 49]], [593, [33, 29]],
  [601, [43, 35]], [617, [29, 27]], [641, [49, 39]], ...
a(4) = -(3*3 - 4*7) = 28 - 9 = 19.

Prog

(PARI) apply( {A254936(n, p=A007519(n))=n=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-3, 4]~}, [1..77]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

Crossrefs

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

Cf. A254937 (corresponding y2-values), A254934 (x1 values), A254935 (y1 values), A255233 (same for primes == 7 mod 8), A255247.

Sequence in context: A291350 A289697 A104296 * A117675 A027726 A395116

Adjacent sequences: A254933 A254934 A254935 * A254937 A254938 A254939

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 18 2015

Extensions

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

Status

approved