- Let
*x*be divisible by*p*then the power of*p*that divides*x*^{n}/*n*is at least*n*- [log_{p}(*n*)], where the latter term denotes the integral part of log_{p}(*n*). In particular, this goes to infinity with*n*. - Let
*x*be divisible by*p*, then the power of*p*that divides*p*^{n}/*n*! is at least*n*- [*n*/*p*^{i}]. In particular, if*p*is odd then the latter term goes to infinity with*n*.

log(1 - *x*) = -

survive in
/
log : *U*_{1}/*p*^{e}

which in fact takes values in the ideal
exp : *p*/*p*^{e}*U*_{1}

by means of the usual power series
exp(*x*) =

which satisfies
exp(