p-adic number

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The p-adic number system for any prime number p extends the ordinary arithmetic
of the rational numbers in a different way from the extension of the rational number system
to the real and complex number systems.

The extension is achieved by an alternative interpretation of the concept of "closeness"
or absolute value.

In particular, two p-adic numbers are considered to be close when
their difference is divisible by a high power of p:
the higher the power, the closer they are.

This property enables p-adic numbers to encode congruence information
in a way that turns out to have powerful applications in number theory –
including, for example, in the famous proof of Fermat's Last Theorem
by Andrew Wiles.[1]

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p-adic analysis is a branch of number theory that deals with
the mathematical analysis of functions of p-adic numbers.

The theory of complex-valued numerical functions on the p-adic numbers is
part of the theory of locally compact groups.
The usual meaning taken for p-adic analysis is the theory of p-adic-valued functions
on spaces of interest.

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