Mysterion

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.

Mysterion