A306618 x-value of the smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4), p = A002145(n).

1, 2, 7, 3, 78, 4, 51, 732, 277, 191, 6, 44, 20621, 122, 416941, 8, 5123, 25, 1034, 9, 3993882, 210107, 203100, 10, 1325, 5248, 65030839, 20107956, 30953, 4584105462, 1036, 4889, 295081, 58746, 20725, 98465863939, 1494439626, 1612, 10173, 6040149252, 102607, 9460742124

Offset

1,2

Comments

a(n) exists for all n.

X = 4*a(n)^2 - (-1)^((p+1)/4), Y = 2*a(n)*A306619(n) gives the smallest solution to x^2 - 2p*y^2 = 1, p = A002145(n).

Links

Table of n, a(n) for n=1..42.

Formula

If the continued fraction of sqrt(2*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/2 and A306619(n) = y.

Example

The smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4) for the first primes congruent to 3 modulo 4:
  n |       Equation      | x_min | y_min
  1 | 2*x^2 -  3*y^2 = -1 |     1 |     1
  2 | 2*x^2 -  7*y^2 = +1 |     2 |     1
  3 | 2*x^2 - 11*y^2 = -1 |     7 |     3
  4 | 2*x^2 - 19*y^2 = -1 |     3 |     1
  5 | 2*x^2 - 23*y^2 = +1 |    78 |    23
  6 | 2*x^2 - 31*y^2 = +1 |     4 |     1
  7 | 2*x^2 - 43*y^2 = -1 |    51 |    11
  8 | 2*x^2 - 47*y^2 = +1 |   732 |   151
  9 | 2*x^2 - 59*y^2 = -1 |   277 |    51

Prog

(PARI) b(p) = if(isprime(p)&&p%4==3, x=1; while(!issquare((2*x^2 - (-1)^((p+1)/4))/p), x++); x)
forprime(p=3, 250, if(p%4==3, print1(b(p), ", ")))

Crossrefs

Cf. A002145, A306619 (y-values).

Similar sequences: A094048, A094049 (x^2 - A002144(n)*y^2 = -1); A306529, A306566 (x^2 - A002145(n)*y^2 = 2*(-1)^((p+1)/4))).

Sequence in context: A090276 A249782 A090564 * A349856 A079072 A004584

Adjacent sequences: A306615 A306616 A306617 * A306619 A306620 A306621

Keyword

nonn

Author

Jianing Song, Mar 25 2019

Status

approved