My goal was to prove the Fundamental Theorem of Arithmetic without using Euclid's Lemma. There are some proofs online but I haven't found one that uses this idea, so I want to make sure it's right.
Please make me aware of any mistakes, I've only be writing proofs for a semester so you would be helping me.
I have already proven, and will be using the fact that the smallest divisor of any integer $\neq$ 1 is prime.
Lemma: The smallest divisor of some number is unique. If it were not, and there were 2 distinct smallest divisors $p,q~,~p\neq q$, we have $\lvert p\rvert\neq\lvert q \rvert$. One of them is smaller than the other so one of them is not the smallest divisor.
Proof: Every prime number has a unique prime factorization (itself). Begin with the smallest composite number 4, the prime factorization of 4 is uniquely $2\cdot2$. Now suppose that every composite number up to but not including $n$ has a unique prime factorization. $n$ has a unique smallest divisor, which is prime, call it $p$. We can uniquely write $n$ as $p\cdot t$. If $t$ is prime then we have a unique prime factorization for $n$. If $t$ is composite, because it is less than $n$, $t$ has a unique prime factorization, and therefore so does $n$.
Applying this reasoning inductively, every composite number has a unique prime factorization. Every prime number also has a unique prime factorization, and so every number has a unique prime factorization.
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