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A306618
x-value of the smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4), p = A002145(n).
3
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).
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
KEYWORD
nonn
AUTHOR
Jianing Song, Mar 25 2019
STATUS
approved