What number does equal its square? Thus, what is the solution of the equation

You would answer they are 0 and 1! You may complain of such trivial question, but some peculiar ecology is existing in finite-digit systems sometimes.

Suppose that we limit the number of digits into 5 and that the equivalence is defined by the 5 digits in our number system. In this number system, we have another solution to (1) which is

Surely,

since the underlined (overflowed) part 8128 is neglected in our digit system
( 5-digit decimal system).
This number is a part of well-known one by recreational
mathematics mania.
Indeed, we can extend the number into any *n*-digit system
as

The question here is how we can derive such *n*-digit numbers in arbitrary
base *p* (*p*-adic number system).
We will discuss other extensions (such that a number equals its cube, etc)
in other articles.