  The issue of which stars may reach the conditions of electron/positron pair
formation instability is of importance to understand the final evolution both
of the first stars and of contemporary stars. The criterion to enter the pair
instability regime in density and temperature is basically controlled by the
mass of the oxygen core. The main sequence masses that produce a given oxygen
core mass are, in turn, dependent on metallicity, mass loss, and convective and
rotationally-induced mixing. We examine the evolution of massive stars to
determine the minimum main sequence mass that can encounter pair-instability
effects, either a pulsational pair instability (PPISN) or a full-fledged
pair-instability supernova (PISN). We concentrate on zero-metallicity stars
with no mass loss subject to the Schwarzschild criterion for convective
instability, but also explore solar metallicity and mass loss and the Ledoux
criterion. As expected, for sufficiently strong rotationally-induced mixing,
the minimum main sequence mass is encountered for conditions that induce
effectively homogeneous evolution such that the original mass is converted
almost entirely to helium and then to oxygen. For this case, we find that the
minimum main sequence mass is ~40 Msun to encounter PPISN and ~65 Msun to
encounter a PISN. When mass-loss is taken into account those mass limits become
~50 Msun for PPISN and ~80 Msun for PISN progenitors. The implications of these
results for the first stars and for contemporary supernovae is discussed.
