Conjecture on combinate of positive integers in terms of primes

Question

Along a heuristic calculation, I am struggeling with a possible proof for my following conjecture:

Every positive integer $n\in \Bbb N$ can be written as a unique combination of $a,b \in \Bbb N$, $m\in \Bbb N_0$ and $p \in \Bbb P$ (a prime), such that:

$$n=a \,p^{m+1}-b\,p^m$$

Has anyone heard yet about such a problem? What might be the proof?

Answer 1

Counterexamples to uniqueness are not hard to find. For example, $36=(2)(3^3)-(2)(3^2)$, and $36=(5)(2^3)-(1)(2^2)$.

Already even if we stick to the single prime $3$, we can express $36$ in infinitely many ways as $(a)(3^3)-(b)(3^2)$.

Answer 2

$14 = 49-5\cdot 7 = 4\cdot4-1\cdot2$.

In particular, this can be done for any prime that divides our starting number.

Answer 3

$$n=a \,p^{m+1}-b\,p^m \Leftrightarrow n=p^m (ap-b) \,.$$

Thus, given any $n>2$ pick a prime $p|n$. Pick $m$ so that $p^m|n$ and $p^{m+1} \nmid n$.

Pick any $a$ positive integer so that $ap > \frac{n}{p^m}$ and let $b= ap- \frac{n}{p^m}$. Then

$$n=p^m(ap-b) \,.$$

It is not hard to prove that, the condition $p^{m+1} \nmid n$ is not needed, and if dropped this method also generates all solutions.