My teacher asked us this question just as a thought, but I don't understand how to even approach it.
If I were to have an equation and proved P(1), P(2) and $\forall n, \in $ $\mathbb N$ $P(n) \rightarrow P(2*n)$ are true, then would $\forall n, \in $ $\mathbb N$ $P(n)$ be true?
I know the usual step would be to prove $P(n) \rightarrow P(n+1) $. Would that mean that you still wouldn't have proven that step, and thus it would be false.
I was also wondering if there was a way to come up with examples disproving them if they were false. I can't even think of an equation that fits the requirement.
Why would $P(3)$ be forced to be true?
You can have other induction patterns. Maybe there are different proofs for odd and even numbers. You might prove $P(1), P(2), P(n) \implies P(n+2)$, which would suffice. I vaguely remember one that proved $P(1), P(2^n)\implies P(2^{n+1})$ and then stepped downward from $P(2^{n+1})$ to $P(2^n+1)$