THEOREM:
1. If a table of pure cubic fields contains all normalized radicands R < U,
then it contains the complete families of radicands associated with all conductors f < U^{2/3}.
2. If a table of pure cubic fields contains the complete families of all conductors f < U,
then it must contain all normalized radicands R < U/3.
PROOF: 1. We estimate the biggest normalized radicand R = m*n^2 associated with a given conductor f = [3*]m*n below the bound B, f < B. (Recall that the radicand R = m*n^2 is normalized, if its square part n is smaller than its linear part m, n < m.) We obtain a chain of inequalities n*n < m*n <= [3*]m*n < B, whence n < B^{1/2}. Consequently the radicand R = m*n^2 = (m*n)*n < B*B^{1/2} = B^{3/2} cannot exceed B^{3/2} (a value that is almost reached for radicands with big square part n approximately equal to m). We conclude that if a table contains all normalized radicands R < U = B^{3/2}, then it contains the complete families of radicands associated with all conductors f < B = U^{2/3}. QED. 2. We give an estimate of the conductor f = [3*]m*n, which belongs to a given normalized radicand R = m*n^2 below the bound B, R < B. Here we have simply the inequality f <= 3*m*n <= 3*m*n^2 = 3*R < 3*B. (The bound 3*B for the conductor is reached, when R is squarefree with n = 1 and of Dedekind type 1.) We conclude that if a table contains the complete families of all conductors f < U = 3*B, then it must contain all normalized radicands R < B = U/3. QED. |