  M dwarf stars are known for their vigorous flaring. This flaring could impact
the climate of orbiting planets, making it important to characterize M dwarf
flares at the short wavelengths that drive atmospheric chemistry and escape. We
conducted a far-ultraviolet flare survey of 6 M dwarfs from the recent MUSCLES
(Measurements of the Ultraviolet Spectral Characteristics of Low-mass
Exoplanetary Systems) observations, as well as 4 highly-active M dwarfs with
archival data. When comparing absolute flare energies, we found the
active-M-star flares to be about 10$\times$ more energetic than inactive-M-star
flares. However, when flare energies were normalized by the star's quiescent
flux, the active and inactive samples exhibited identical flare distributions,
with a power-law index of -$0.76^{+0.1}_{-0.09}$ (cumulative distribution). The
rate and distribution of flares are such that they could dominate the FUV
energy budget of M dwarfs, assuming the same distribution holds to flares as
energetic as those cataloged by Kepler and ground-based surveys. We used the
observed events to create an idealized model flare with realistic spectral and
temporal energy budgets to be used in photochemical simulations of exoplanet
atmospheres. Applied to our own simulation of direct photolysis by photons
alone (no particles), we find the most energetic observed flares have little
effect on an Earth-like atmosphere, photolyzing $\sim$0.01% of the total O$_3$
column. The observations were too limited temporally (73 h cumulative exposure)
to catch rare, highly energetic flares. Those that the power-law fit predicts
occur monthly would photolyze $\sim$1% of the O$_3$ column and those it
predicts occur yearly would photolyze the full O$_3$ column. Whether such
energetic flares occur at the rate predicted is an open question.
