Definition
Given integer $a$ and $b$, with $a>1$, there exist integer $r$
Let W be the function , function define as follows
$$W(a,b)=r$$
Where $$r=r_1+r_2+…+r_{m+1}$$
And
$$a\cdot q_1=b+r_1$$
$$a\cdot q_2=q_1+r_2$$
$$a\cdot q_3=q_2+r_3$$
$$\vdots$$
$$a\cdot q_{m+1}=q_m+r_{m+1}=a$$
Up to $q_{m+1}=1$
And
$0\leq r_i<a$ for $i={1,2,…,m+1}$.
More simply
$a^{m+1}=b+r_1+ar_2+a^2r_3+…+a^mr_{m+1}$
For example
$W(5,17)=4$.
Here are some More interesting properties which I already proved
$r+b=1 \mod a-1$
$W(odd,odd)=even$
$W(odd,even)=odd$
I want more information on such type of function/algorithm. Properties of this type of function which deep connection with number theory. its really helpfull for me.
Best Answer
This answer assumes $b>0$. Also, while this does not give useful references, it does map the original problem into a very well-known related problem.
Take $m$ to be the smallest integer s.t. $a^{m+1} \ge b$, i.e. $m = \lceil \log{b}/\log{a} \rceil - 1$.
As you said, $a^{m+1}=b+r_1+ar_2+a^2r_3+...+a^mr_{m+1}$.
Rearranging, we get: $(a^{m+1} - b) = r_{m+1} a^m + ... + r_3 a^2 + r_2 a^1 + r_1 a^0 $.
In other words, the sequence of $m+1$ digits $(r_{m+1}, ..., r_3, r_2, r_1)$ is the representation of the non-negative integer $(a^{m+1} - b)$ in base $a$, and so $W(a,b) =r=$ the digit sum in this base-$a$ representation.
E.g. for $a=5, b=17$ we have:
$m+1 = 2$ since $5^2 > 17$
$a^{m+1} - b = 5^2 - 17 = 25-17 = 8 = 13_5$ (i.e. $13$ in base $5$).
Sum of digits $W(5,17) = r = 1+3 = 4$ as desired.
E.g. proof that $b + r = 1 \mod (a-1)$:
$a^k \mod (a-1) = 1^k \mod (a-1) = 1$ for any $k$
So $\mod (a-1)$ we have: $r = \sum r_k = r_{m+1} a^m + ... + r_3 a^2 + r_2 a^1 + r_1 a^0 = (a^{m+1} - b) = 1 - b$.
You can probably prove many more interesting things about $W(a,b)$ based on understanding it as the digit sum of $(a^{m+1} - b)$ in base $a$.