Conjecture on combinate of positive integers in terms of primes 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?

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 $(a)(3^3)-(b)(3^2)$.