I'm striving for proper and standard mathematical writing, elegance being a plus.
Here's what I want to say. If two natural numbers are not the same number, the fundamental theorem of arithmetic implies that, comparing both numbers' prime factorizations, there must be at least one prime $p_i$ whose power is not the same in both numbers. (Otherwise, they'd be the same natural number.) (There's some problems with saying it like, but I'll certainly get what I mean if you continue to read.)
I'd like to be able to say this in a brief manner, so I'm willing to spend a Lemma or a notation note to clarify this fact.
Here's some technical things that are of concern. For example, two numbers don't always have the same primes, of course. The obvious solution I found is to say that we can pair-up two factorizations by making sure we express both numbers with the same number of prime factors, but giving primes an exponent of $0$ if that prime doesn't belong in the number's factorization.
Example. Let $N = 2\cdot3^2\cdot5^3$, $M = 2\cdot3^3\cdot7^4$. We can express them as \begin{align} N &= 2 \cdot 3^2 \cdot 5^3 \cdot 7^0\\ M &= 2 \cdot 3^3 \cdot 5^0 \cdot 7^4 \end{align}
Now I could say, of two naturals $X$, $Y$ in general,
\begin{align} X &= p_1^{e'_1} \cdot p_2^{e'_2} \cdots p_i^{e'_i} \cdots p_k^{e'_k}\\ Y &= p_1^{e''_1} \cdot p_2^{e''_2} \cdots p_i^{e''_i} \cdots p_k^{e''_k}, \end{align}
for some $k$ natural. If $X \ne Y$, there must be some index $j \in \{1, 2, .., k\}$ such that $e'_j \ne e''_j$ --- otherwise $X = Y$.
I don't like the apostrophes in the exponents. What I actually prefer to do is to turn $e$ into a function of two arguments, so $e(i,N)$ represents the exponent of prime $i$ in number $N$. (So $e(3,M) = 0$ since $5^0$ is the third prime in $M$'s expression in the example above, $e(2,N) = 2$ since the second prime in in $N$'s expression, ...) It takes more space, but I think it's much clearer.
Of course that writing primes with a zero exponent is not given by the fundamental theorem, but I thought it simplified my objective here.
What would really score an extra-credit in your answer is if you cite a reference that applies your suggestion --- in particular if it's a book about style in mathematical writing.
If you're striving for elegance, you should prove the contrapositive of the statement: if two numbers have identical prime factorizations, then they are the same number. It'll be easy to find samples of that proof, since it is the uniqueness claim of the Fundamental Theorem of Arithmetic.