A306619 y-value of the smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4), p = A002145(n).

1, 1, 3, 1, 23, 1, 11, 151, 51, 33, 1, 7, 3201, 17, 57003, 1, 633, 3, 119, 1, 437071, 22209, 20783, 1, 129, 497, 6104097, 1839433, 399752993, 89, 411, 23817, 4711, 1611, 7475426163, 111543983, 119, 739, 436478927, 7089, 644468311, 103, 93487270491, 573497, 57, 4182991

Offset

1,3

Comments

a(n) exists for all n.

X = 4*A306618(n)^2 - (-1)^((p+1)/4), Y = 2*A306618(n)*a(n) gives the smallest solution to x^2 - 2p*y^2 = 1, p = A002145(n).

Links

Table of n, a(n) for n=1..46.

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 A306618(n) = x/2 and a(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, y=1; while(!issquare((p*y^2 + (-1)^((p+1)/4))/2), y++); y)
forprime(p=3, 250, if(p%4==3, print1(b(p), ", ")))

Crossrefs

Cf. A002145, A306618 (x-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: A010291 A372167 A370948 * A335644 A027477 A260780

Adjacent sequences: A306616 A306617 A306618 * A306620 A306621 A306622

Keyword

nonn

Author

Jianing Song, Mar 25 2019

Status

approved