A306529 x-value of the smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4), p = A002145(n).

1, 3, 3, 13, 5, 39, 59, 7, 23, 221, 59, 9, 9, 477, 31, 2175, 103, 8807, 41571, 8005, 13, 2047, 2999, 127539, 527593, 15, 15, 2489, 1917, 373, 340551, 11759, 9409, 4109, 52778687, 801, 19, 137913, 113759383, 137, 16437, 12311, 21, 21, 15732537, 1275, 1729, 7204587, 305987, 67

Offset

1,2

Comments

a(n) exists for all n.

X = a(n)^2 - (-1)^((p+1)/4), Y = a(n)*A306566(n) gives the smallest solution to x^2 - p*y^2 = 1, p = A002145(n). As a result, all the positive solutions to x^2 - p*y^2 = 2*(-1)^((p+1)/4) are given by (x_n, y_n) where x_n + (y_n)*sqrt(p) = (a(n) + A306566(n)*sqrt(p))*(X + Y*sqrt(p))^n.

Links

Table of n, a(n) for n=1..50.

Formula

If the continued fraction of sqrt(A002145(n)) is [a_0; {a_1, a_2, ..., a_(k-1), a_k, a_(k-1), ..., a_1, 2*a_0}], where {} is the periodic part, let x/y = [a_0; a_1, a_2, ..., a_(k-1)], gcd(x, y) = 1, then a(n) = x and A306566(n) = y.

Example

The smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4) for the first primes congruent to 3 modulo 4:
  n |      Equation     | x_min | y_min
  1 | x^2 -  3*y^2 = -2 |     1 |     1
  2 | x^2 -  7*y^2 = +2 |     3 |     1
  3 | x^2 - 11*y^2 = -2 |     3 |     1
  4 | x^2 - 19*y^2 = -2 |    13 |     3
  5 | x^2 - 23*y^2 = +2 |     5 |     1
  6 | x^2 - 31*y^2 = +2 |    39 |     7
  7 | x^2 - 43*y^2 = -2 |    59 |     9
  8 | x^2 - 47*y^2 = +2 |     7 |     1
  9 | x^2 - 59*y^2 = -2 |    23 |     3

Prog

(PARI) b(p) = if(isprime(p)&&p%4==3, x=1; while(!issquare((x^2 - 2*(-1)^((p+1)/4))/p), x++); x)
forprime(p=3, 500, if(p%4==3, print1(b(p), ", ")))

Crossrefs

Cf. A002145, A306566 (y-values).

Similar sequences: A094048, A094049 (x^2 - A002144(n)*y^2 = -1); A306618, A306619 (2*x^2 - A002145(n)*y^2 = (-1)^((p+1)/4))).

Sequence in context: A206704 A020550 A368960 * A200921 A135949 A168234

Adjacent sequences: A306526 A306527 A306528 * A306530 A306531 A306532

Keyword

nonn

Author

Jianing Song, Mar 25 2019

Status

approved