Invariance of number properties under different bases Are in number theory always numbers with basis 10 considered? I‘m asking which role the basis plays in number theoretic properties or notions like prime numbers for examples.
For example: the number 3 is a prime number.
When I take know the basis 3 to represent 3, then it is 10 what is not prime. However, 10 is still in $\mathbb Z$.
Knows somebody if bases play a role in theory of numbers? And if yes, are there invariant properties?
(Remark:
Basis 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …
Basis  3 : 0, 1, 2, 10, 11, 12, 20, 21, 22, 100, …)
 A: This is an interesting question, because it addresses the difference between what a number is and how we choose to name it or write it.
The properties of numbers that matter in mathematics (even? prime? square?) belong to the numbers themselves. They are the same whether you are speaking about them in English or in French, whether you write them in base 10 or base 2 or using Roman numerals.
There are some properties that do depend on the representations in particular bases. These are often part of "recreational mathematics" - puzzles or observations that are fun to think about but don't often make their way into "serious" mathematics.  For example, $14641$ is the fourth power of some number. Find it. Then figure out when the number with those digits thought of as in some other base $b$ is also a fourth power, and why.

 Prove that the number 14641 is the fourth power of an integer in any base greater than 6?

A: Most properties such as being prime or being a perfect square stay invariant in any base. If $n$ divides $m$ in one base, it will divide $m$  in any other base as well.
The number of digits and the digits which are allowed depends on the base.
Numbers are the same, using different bases changes the way we express that number.
