Are numbers independent of the basis we use?

Question

So I have recently read about Kaprekar's Constant (https://en.wikipedia.org/wiki/6174_(number)) and It made me wonder If this number is really "special"? It seems to me that the notion 6174 (and the proof of its apparent meaning)is highly dependent on the number base that we use, If that is true than it is not so special because it means we have deliberately cooked up that number system to obtain that "special" number. Is this really the case. Can't we use generalised numbers while doing number theory like we use tensors independent of coordinate systems?

And if they are really independent of the basis we use can we transform special numbers from basis to basis, preserving its unique attribute?

Answer 1

When we change base then the digits will change accordingly, therefore properties which depend on digits will be altered.

For example in base $8$ we have $$3\times 3 = 11$$ The sum of digits of $11$ is $2$ which is not a multiple of $3$

As you know in base $10$ if a number is a multiple of $3$, the sum of its digits is also a multiple of $3$ a property which is obviously lost in base $8$

Answer 2

Most of the corpus of the theory of numbers relies on the abstract definition of integers and is completely representation-independent.

For example, $x^n+y^n=z^n$ has no solution for all $n>2$, regardless the basis.

Properties that refer to base $10$ (or another one) are often more of an anecdotical character.

---

Anyway, the digital representation of numbers are in fact polynomials of some $b$ in $\mathbb Z_b$ and can be studied for their general properties.

Another important representation is the prime factor decomposition, that lists the multiplicities of the prime factors.

E.g.

$$6174=2\cdot3^2\cdot7^3\leftrightarrow 1,2,0,3.$$