  We provide explicit formulas for the number of vacua of four-dimensional pure
N=1 super Yang-Mills theories on a circle, with any simple gauge algebra and
any choice of center and spectrum of line operators. These form a key
ingredient in the semi-classical calculation of the number of massive vacua of
N=1* gauge theories with gauge algebra su(n) compactified on a circle. Using
arithmetic, we express that number in an SL(2,Z) duality invariant manner. We
confirm our tally of massive vacua of the N=1* theories by a count of
inequivalent extrema of the exact superpotential. Furthermore, we compute a
formula for a refined index that distinguishes massive vacua according to their
unbroken discrete gauge group.
