Theorems like (factorization theorem) aren't suitable to be proven with induction since induction requires that the property tackled with our set of variables must be satisfied consecutively; (prime numbers aren't consecutive). So, how could we be sure that any theorem that we want to prove by induction, would be suitable for that? Is the answer: If it was proven as a false theorem by induction?
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1Actually, the fundamental theorem of arithmetic is often proven using induction. – noedne Apr 12 '18 at 09:07
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https://www.youtube.com/watch?v=cm0io4R0tOM at 22:40 this is what I meant when I talked about prime numbers – Apr 12 '18 at 09:28
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The trivial answer is that statements of the form $$\forall n \in \mathbb{N}: I(n),$$ where $I(n)$ is any statement, are the things to prove via induction. But there are of course more possibilities. – Luke Apr 12 '18 at 10:02