OFFSET
1,1
COMMENTS
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).
FORMULA
CROSSREFS
KEYWORD
nonn,uned,more
AUTHOR
Vladimir Shevelev, Aug 28 2010
STATUS
approved