A377600 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = -3 is solvable over the integers.

1, 3, 4, 7, 12, 13, 19, 21, 28, 31, 39, 43, 52, 57, 61, 67, 73, 76, 84, 91, 93, 97, 103, 109, 111, 124, 127, 129, 133, 139, 147, 151, 157, 163, 172, 181, 183, 193, 199, 201, 211, 217, 228, 237, 241, 244, 247, 259, 268, 271, 273, 277, 283, 292, 301, 307, 309, 313, 327, 331, 337, 343, 364

Offset

1,2

Comments

Calculated using Dario Alpern's quadratic Diophantine solver, see link.

Links

Robin Visser, Table of n, a(n) for n = 1..10000

Dario Alpern, Generic two integer variable equation solver.

Eric Weisstein's World of Mathematics, Pell Equation.

Example

The first fundamental solutions [x(n), y(n)] are (the first entry gives D(n)=a(n)):
[1, [1, 2]], [3, [0, 1]], [4, [1, 1]], [7, [2, 1]], [12, [3, 1]], [13, [7, 2]], [19, [4, 1]], [21, [9, 2]], [28, [5, 1]], [31, [11, 2]], [39, [6, 1]], [43, [13, 2]], [52, [7, 1]], [57, [15, 2]], [61, [5639, 722]], [67, [8, 1]], [73, [17, 2]], [76, [61, 7]], [84, [9, 1]], [91, [19, 2]], [93, [135, 14]], [97, [847, 86]], [103, [10, 1]], [109, [1399, 134]], [111, [21, 2]], [124, [11, 1]], [127, [293, 26]], [129, [159, 14]], [133, [23, 2]], [139, [224, 19]], [147, [12, 1]], [151, [86, 7]], [157, [25, 2]], [163, [932, 73]], [172, [13, 1]], [181, [11262809, 837158]], [183, [27, 2]], [193, [189743, 13658]], [199, [14, 1]], ...

Prog

(Python)
from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
def A377600_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda d:len(diop_DN(d, -3)), count(max(startvalue, 1)))
A377600_list = list(islice(A377600_gen(), 63)) # Chai Wah Wu, Nov 03 2024

Crossrefs

Cf. A031396, A244819, A261246, A377607.

Sequence in context: A286728 A300332 A244819 * A305185 A083561 A018195

Adjacent sequences: A377597 A377598 A377599 * A377601 A377602 A377603

Keyword

nonn

Author

Robin Visser, Nov 02 2024

Status

approved