  We describe two new packages ExactpAdics and ExactpAdicsII for the Magma
computer algebra system for working with p-adic numbers exactly, in the sense
that numbers are represented lazily to infinite p-adic precision. This has the
benefits of increasing user-friendliness and speeding up some computations, as
well as forcibly producing provable results. The two packages use different
methods for lazy evaluation, which we describe and compare in detail. The
intention is that this article will be of benefit to anyone wanting to
implement similar functionality in other languages.
