Are there any odd perfect numbers in base $g$, where $g \neq 10$?

Question

The topic of odd perfect numbers likely needs no introduction.

Here is my primary question:

Are there any odd perfect numbers in base $g$, where $g \neq 10$?

Of course, there are several subproblems:

(1) How do you define primality in base $g$?

(2) After (1), how do you define the sum of divisors $\sigma$ in base $g$?

(3) Lastly, when do you consider a number in base $g$ as odd or even, particularly when the number $g$ is odd in base $10$?

MY OWN ATTEMPTS AT FINDING OUT ANSWERS TO MY QUESTIONS

The answer to (1) appears to be contained essentially in this Math Forum Q&A.

Given the answer to (1), and since we expect the sum of divisors function $\sigma$ in (2) to also be multiplicative, it follows that the usual sum of divisors function in base $10$ can also be considered as the sum of divisors function in any given base $g \neq 10$.

Lastly, I currently do not know of a complete answer to my question (3). Hence, I do not know of any odd perfect numbers in base $g$, where $g \neq 10$.

Answer 1

Being odd or being perfect for some integer $n$ is not at all related to what base we are working over for number notation. (it's the essence of the answer in your linked Q&A as well, so you should have known).

So it's still open.

$n$ is an odd integer means that $n = 2m+1$ for some integer $m$ (here with integers I mean $\Bbb N=\{0,1,2,3,\ldots\}$) (so we only need to know what $2=1+1$ is and what multiplication and addition are). $d$ is a divisor of $n$ iff there is some integer $k$ with $k\times d = n$ (only need to know multiplication) and $n$ is perfect iff $n = \sum \{d: d \text{ a divisor of } n, d \neq n\}$. Addition and multiplication are intrinsic (not depending on any base representation) so these definitions are intrinsic too.

Answer 2

(1) same way you do in any other base, does it divide by any other number less than it and is it greater than 1.

(2) The same way you can in any other base ( representations of divisors will be different, but the final representation of the sum of them will still be equal when transformed across bases)

(3) in an odd base, an odd number has to have an odd number of odd digits. An even number would have an even number of odd digits (because the definition of even or odd is still divisibility by 2 so base independent only representation is affected). In an even base it all falls to the last digit being odd or even.