OFFSET
1,4
COMMENTS
Let F_k(X,Y) denote the polynomial X^3 - k*X^2*Y - (k+3)*X*Y^2 - Y^3. Then if there exist integers X,Y such that F_k(X,Y) is a divisor of k^2 + 3*k + 9, then F_{-k-3}(-Y, -X) is also a divisor of (-k-3)^2 + 3*(-k-3) + 9. Thus, to classify all integers k such that F_k(X,Y) divides k^2 + 3*k + 9 for some integers X and Y, it suffices to classify only those integers k where k >= -1.
Let L_k denote Shanks' simplest cubic field with defining polynomial x^3 - k*x^2 - (k+3)*x - 1. Then this sequence is also the set of all integers k >= -1 such that L_k is isomorphic to L_i for some i >= -1 not equal to k (as shown by Hoshi and Okazaki). In particular, we have the following isomorphisms between the simplest cubic fields: L_{-1} = L_5 = L_12 = L_1259, L_0 = L_3 = L_54, L_1 = L_66, and L_2 = L_2389.
LINKS
A. Hoshi, On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields, J. Number Theory 131 (2011), no. 11, 2135-2150. See Table 1 on page 2138.
T. Kashio and R. Sekigawa, The characterization of cyclic cubic fields with power integral bases, Kodai Math. J. 44 (2021), no. 2, 290-306.
R. Okazaki, The simplest cubic fields are non-isomorphic to each other, Doshisha University, Kyoto, Japan. [Cached copy at the Wayback Machine]
M. Tinková, R. Visser, and P. Yatsyna, Sums of two units in number fields, arXiv:2502.01345 [math.NT], 2025. See page 9.
EXAMPLE
For each integer k >= -1 given in the sequence, the list of all pairs (X,Y) such that X*Y*(X+Y) is nonzero and such that F_k(X,Y) is a positive integer divisor of k^2 + 3k + 9 is as follows:
If k = -1, then (X,Y) = (-1,-1), (-1,2), (2,-1), (5,4), (4,-9), (-9,5), (2,1), (1,-3), or (-3,2).
If k = 0, then (X,Y) = (2,1), (1,-3), (-3, 2), (-1,-1), (-1, 2), or (2,-1).
If k = 1, then (X,Y) = (-5,-2), (-2, 7), or (7,-5).
If k = 2, then (X,Y) = (-7,-2), (-2, 9), or (9,-7).
If k = 3, then (X,Y) = (-1,-1), (-1, 2), (2,-1), (-4,-1), (-1, 5), or (5,-4).
If k = 5, then (X,Y) = (-1,-2), (-2, 3), (3,-1), (-4,-1), (-1, 5), (5,-4), (19, 3), (3,-22), or (-22, 19).
If k = 12, then (X,Y) = (-1,-1), (-1, 2), (2,-1), (-13,-1), (-1, 14), (14,-13), (-4,-1), (-1, 5), or (5,-4).
If k = 54, then (X,Y) = (-1,-2), (-2, 3), (3,-1), (-4,-1), (-1, 5), or (5,-4).
If k = 66, then (X,Y) = (-5,-2), (-2, 7), or (7,-5).
If k = 1259, then (X,Y) = (-4,-5), (-5, 9), (9,-4), (-13,-1), (-1, 14), (14,-13), (-3,-19), (-19, 22), or (22,-3).
If k = 2389, then (X,Y) = (-7,-2), (-2, 9), or (9,-7).
PROG
(Magma)
is_A383068 := function(k)
R<x> := PolynomialRing(Integers());
T := Thue(x^3 - k*x^2 - (k+3)*x - 1);
for d in Divisors(k^2 + 3*k + 9) do
S := Solutions(T, d);
for s in S do
if (s[1]*s[2]*(s[1]+s[2]) ne 0) then return true; end if;
end for;
end for;
return false;
end function;
[k : k in [-1..3000] | is_A383068(k)];
CROSSREFS
KEYWORD
sign,fini,full
AUTHOR
Robin Visser, Apr 15 2025
STATUS
approved
