I am trying to understand a paper where a numerical algorithm is described. I do not understand the point where the expression "exact power of a prime that divides a number" is used. Here is the text with all related context and the exact point in bold:
Let $n_0$ be a fixed (small) positive integer, and $n≥4n_0$ (...)
Let $m$ be a positive integer
- Compute the prime factors $p_1$, . . . , $p_l$ of $m$
- For index $j$, $1≤j≤k$, perform the following operations :
- Assign $a = n − j + 1$ and $b =j$.
- Decompose $a$ and $b$ with the powers of $p_i$, in the form $$a=a^{*}×p_1^{α_1}...p_l^{α_l}, b=b^{∗}×p_1^{β_1}...p_l^{β_l}$$ For each $i$, $p_i^{α_i}$ is the exact power of $p_i$ that divides $a$, $p_i^{β_i}$ is the exact power of $p_i$ that divides $b$, so that $a^∗$ and $b^∗$ do not have one of the $p_i$ as a prime factor.
What I am struggling to understand is that the $p$ prime number is a factor of $m$ but not necessarily of $a$. The definition of $p_i^{\alpha_i}$ calls for an exact division of $a$, not the largest power that is not greater than $a$.
For instance
$$ n_0 = 2 \rightarrow n = 4n_0 = 8 \\ m = 21 \rightarrow p_1 = 3; p_2 = 7 \\ j = 1 \rightarrow a = 8 - 1 + 1 = 8; b = 1 \\ p_1^{\alpha_1} = 8 / 3^{\alpha} $$
where the last expression does not seem to have an exact (integer) solution.
Am I misunderstanding the definition of "exact division"? Or am I missing something in the algorithm steps leading to this point?
Any help is appreciated.
The exact power here means the maximal power that divides the number.
So the exact power of $5$ that divides $50$ is $2$ as $5^2$ divides $50$ while $5^3$ does not.
If a primes does not divide a number the exact power that divides it is the $0$-th power.