Basic p-adic Arithmetic

Addition and multiplication of p-adic numbers are defined in several cases whether or not both numbers are given in Hensel expansion. Note that addition and multiplication in can be defined by

The left-hand expansions are Hensel expansions, but the right-hand ones, in general, are not.

Also p-adic exponentiation is implemented for the case of integer powers. We then define the p-adic version of some classical functions, such as log, exp, sin, cos, sinh, and cosh. This is done with the help of their power series expansions, which are completely analogous to the real case except that their regions of convergence might differ [1]. We have also defined square roots. The existence of square roots of a number in depends on the following theorems.

Let . A p-adic number is a square if and only if is a square residue mod .

Let . A 2-adic number is a square if and only if .

Finally, we implement the p-adic Gamma function.

Here are some examples.