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A383068 The set of all integers k >= -1 with the property that there exist integers X and Y such that X*Y*(X+Y) is nonzero and X^3 - k*X^2*Y - (k+3)*X*Y^2 - Y^3 is a divisor of k^2 + 3*k + 9.

%I A383068 #9 Apr 19 2025 06:16:00
%S A383068 -1,0,1,2,3,5,12,54,66,1259,2389
%N A383068 The set of all integers k >= -1 with the property that there exist integers X and Y such that X*Y*(X+Y) is nonzero and X^3 - k*X^2*Y - (k+3)*X*Y^2 - Y^3 is a divisor of k^2 + 3*k + 9.
%C A383068 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.
%C A383068 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.
%H A383068 A. Hoshi, <a href="https://doi.org/10.1016/j.jnt.2011.05.001">On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields</a>, J. Number Theory 131 (2011), no. 11, 2135-2150.  See Table 1 on page 2138.
%H A383068 T. Kashio and R. Sekigawa, <a href="https://doi.org/10.2996/kmj44204">The characterization of cyclic cubic fields with power integral bases</a>, Kodai Math. J. 44 (2021), no. 2, 290-306.
%H A383068 R. Okazaki, <a href="https://web.archive.org/web/20070627235451/http://www1.doshisha.ac.jp/~rokazaki/Simplest.pdf">The simplest cubic fields are non-isomorphic to each other</a>, Doshisha University, Kyoto, Japan. [Cached copy at the Wayback Machine]
%H A383068 M. Tinková, R. Visser, and P. Yatsyna, <a href="https://arxiv.org/abs/2502.01345">Sums of two units in number fields</a>, arXiv:2502.01345 [math.NT], 2025.  See page 9.
%e A383068 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:
%e A383068 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).
%e A383068 If k = 0, then (X,Y) = (2,1), (1,-3), (-3, 2), (-1,-1), (-1, 2), or (2,-1).
%e A383068 If k = 1, then (X,Y) = (-5,-2), (-2, 7), or (7,-5).
%e A383068 If k = 2, then (X,Y) = (-7,-2), (-2, 9), or (9,-7).
%e A383068 If k = 3, then (X,Y) = (-1,-1), (-1, 2), (2,-1), (-4,-1), (-1, 5), or (5,-4).
%e A383068 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).
%e A383068 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).
%e A383068 If k = 54, then (X,Y) = (-1,-2), (-2, 3), (3,-1), (-4,-1), (-1, 5), or (5,-4).
%e A383068 If k = 66, then (X,Y) = (-5,-2), (-2, 7), or (7,-5).
%e A383068 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).
%e A383068 If k = 2389, then (X,Y) = (-7,-2), (-2, 9), or (9,-7).
%o A383068 (Magma)
%o A383068 is_A383068 := function(k)
%o A383068     R<x> := PolynomialRing(Integers());
%o A383068     T := Thue(x^3 - k*x^2 - (k+3)*x - 1);
%o A383068     for d in Divisors(k^2 + 3*k + 9) do
%o A383068         S := Solutions(T, d);
%o A383068         for s in S do
%o A383068             if (s[1]*s[2]*(s[1]+s[2]) ne 0) then return true; end if;
%o A383068         end for;
%o A383068     end for;
%o A383068     return false;
%o A383068 end function;
%o A383068 [k : k in [-1..3000] | is_A383068(k)];
%Y A383068 Cf. A379019, A383067.
%K A383068 sign,fini,full
%O A383068 1,4
%A A383068 _Robin Visser_, Apr 15 2025