  Within the context of massive N-component $\phi^4$ scalar field theory, we
use asymptotic Pade-approximant methods to estimate from prior orders of
perturbation theory the five-loop contributions to the coupling-constant
beta-function $\beta_g$, the anomalous mass dimension $\gamma_m$, the
vacuum-energy beta-function $\beta_v$, and the anomalous dimension $\gamma_2$
of the scalar field propagator. These estimates are then compared with explicit
calculations of the five-loop contributions to $\beta_g$, $\gamma_m$,
$\beta_v$, and are seen to be respectively within 5%, 18%, and 27% of their
true values for $N$ between 1 and 5. We then extend asymptotic Pade-approximant
methods to predict the presently unknown six-loop contributions to $\beta_g$,
$\gamma_m$, and $\beta_v$. These predictions, as well as the six-loop
prediction for $\gamma_2$, provide a test of asymptotic Pade-approximant
methods against future calculations.
