I've been puzzled by a step in D'Aurizio's proof concerning the finiteness of a certain subset $J_p$ of $\mathbf{N}$: $$J_p = \{n : p \text{ divides the numerator of } H_n\}.$$ His paper is here: http://arxiv.org/abs/1102.0765. [Note that he refers to $J_p$ unconventionally as $M_p$; everybody, including all the papers he cites, calls this set $J_p$.]
A lemma from within reads
Lemma 2. If $a$, $b$, $c$ are three distinct elements of $J_p$ with $a < b$ and $c - b \equiv 0$ (mod $p$) then $$c - b + a \in J_p.$$
Immediately after the proof of the lemma, the argument becomes hard to follow:
Let us suppose, now, that $J_p$ is infinite. Then at least one residue class in $J_p/(p^2-p)$ is infinite; let $a$ be the smallest positive integer in such a class
es. $p^2-p$ belongs to $J_p$, so, by Lemma 2, we have that $$a + (p^2-p)\mathbf{N} \subseteq J_p\tag{1}$$ holds, and by applying Lemma 2 again, $$(p^2-p)\mathbf{N}\subseteq J_p.\tag{2}$$
Let's pick this apart. An infinite subset of $\mathbf{Z}$ modulo any nonzero integer will split into finitely many classes, so they can't all be finite. Thus, at least one residue class of $J_p$ modulo $p^2-p$ is infinite: pick one such class and call it $S$. Let $a = \min S$. The fact that $p^2 - p$ belongs to $J_p$ is a theorem of Eswarathasan and Levine. So far, so good.
Questions.
- Why is (1) true?
- How does (2) follow from (1)?