On some subtleties about different principles of mathematical induction? I'm reading Gallian's "Contemporary Abstract Algebra", here there are the following principles of mathematical induction from the aforementioned book:






Now, there is this proof of the fundamental theorem of arithmetic:




And then there is this commentary:



Why is the proof with the second principle of mathematical induction "more natural" than the proof with the first principle of mathematical induction? I don't see what is the difference of the proofs produced with the first and second principles of mathematical induction. It's not clear to me why we can't produce the same proof in there with the first principle of mathematical induction. Can you help?
 A: The proof does not work with the first principle (often called "weak induction", as opposed to "strong induction" for the second principle) because the first principle only assumes the inductive hypothesis for the immediate predecessor.
But this is not enough for the argument given: when we reduce to cases and conclude that either $n+1$ is prime or else $n+1$ factors as $ab$ with $a, b \leq n$, we cannot make further progress on the second case without assuming the inductive hypotheses for all $k \leq n$ (so that, in particular, we can capture $a$ and $b$).
As an aside, it baffles me that so much pedagogical attention is paid to the distinction here, when we can simply always use strong induction and then never have to worry...
A: I suggest you don't memorize these formalities.
I've never heard of these terms "1st principle" and "2nd principle" yet I've read lots of proofs using both forms and I can understand them just fine.
Both forms are natural. It's all about the problem at hand.
Sometimes to prove something about $n+1$, it is just not sufficient to know that that same thing holds for $n$. But in these cases it may turn out that if you assume that thing (statement) holds for all integers k:  $1 \le k \le n$, then you can easily prove it for $n+1$. That's exactly what that  commentary is saying.
That 2nd form may be called strong induction but I think in a way it's weaker. Why? Because your assumption is stronger hence your job when doing the induction step is in fact easier.
But OK, I guess they call it strong induction exactly because the assumption is stronger, not because the process itself is stronger.
You can even have forms of "multidimensional induction" but anyway... I guess that's another topic.
