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

Author

Robin Visser, Nov 02 2024

Keywords

• nonn

Comments

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

Examples

			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]], ...
		

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.

Crossrefs

Cf. A031396, A244819, A261246, A377607.

Programs

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