%N Positive numbers l of the form l = A007913(k^4 - 4*k*m^3), where 1 <= k <= 5*l, 1 <= |m| <= 5*l.
%C The equation x^3 + y^3 + z^3 = 0 is solvable in numbers of the form N + M*sqrt(a(n)), where M and N are integers. Moreover, it is solvable in numbers of the form N + M*sqrt(l), where l has the form l = A007913(k^4 - 4*k*m^3), where k,|m| >= 1 (without restrictions k,|m| <= 5*l). But in this more general case there could be unknown numbers l having this form; this circumstance does not allow construction of the full sequence of such l. Therefore we restrict ourselves by the condition k,|m| <= 5*l. Note that testing l with respect to this condition is rather simple by sorting all values of k,|m| <= l. One can prove that, at least, if the Fermat numbers (A000215) are squarefree, then the sequence is infinite. Conjecture (necessity of the form of l): If the equation x^3 + y^3 + z^3 = 0 is solvable in numbers of the form N + M*sqrt(l) with integer N,M, then there exist positive integers k,m such that l = A007913(k^4 - 4*k*m^3).
%F Let a(n) = A007913(k^4-4*k*m^3). Put g=sqrt(A008833(k^4-4*k*m^3)). Then identity A^3+B^3+C^3=0 is valid, where A=2*m^6-k^3*m^3-k^6+k*(k^3+5*m^3)*g*sqrt(a(n)); B=3*m*(k^3-m^3)*(k^2-g*sqrt(a(n))); C=k^6-8*k^3*m^3-2*m^6-k*(k^3-4*m^3)*g*sqrt(a(n)).
%Y Cf. A007913, A008833, A000215.
%A _Vladimir Shevelev_, Aug 28 2010