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Claim. For a solution $(P,Q)$ of Problem 307, the products $n=\prod P$ and $m=\prod Q$ satisfy $n'=m$ and $m'=n$, where $x\mapsto x'$ is the arithmetic derivative: for squarefree $n$, $n'=n\sum_{p\in P}1/p$, which is the problem page's $N(P)=\prod Q$, and likewise $m'=N(Q)=\prod P$. So a solution is a two-cycle of the arithmetic derivative that is not a fixed point. Kovič (§3.2, pp. 7--9) proves the following of such cycles, after recalling Ufnarovski and Åhlander's result (J. Integer Seq. 6 (2003)) that the two members are squarefree with disjoint sets of prime factors.

• Proposition 16 (p. 8): the system $n'=m$, $m'=n$ has no solution with both $m$ and $n$ products of two primes; so no solution has $|P|=|Q|=2$.
• A computer search (p. 8) found no solution of $x''=x$ with $x<10000$; the fixed points $p^p$, which also satisfy it, are excluded tacitly. So both products of a solution are at least $10^4$.
• Proposition 17 (p. 8): if $n'=m$, $m'=n$ and the smaller member $n<m$ is odd, then $n$ has at least nine prime factors and $n>3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29$. So $2\notin P$ with $\prod P<\prod Q$ forces $|P|\ge9$.
• Proposition 19 (p. 9): if both members are odd, with $r_1$ and $s_1$ the numbers of prime factors of $n$ congruent to $1$ and to $-1$ modulo $4$, and $r_2$, $s_2$ the same for $m$, then $(r_1-s_1)(r_2-s_2)\equiv1\pmod 4$.

Covers. These necessary conditions, each of which refutes every solution it excludes: a partial no to the existence question. Not covered: whether a solution exists. Proposition 18 (p. 8), which asserts $r+s\ge34$ for the numbers $r$ and $s$ of prime factors of $n$ and $m$ (with one member having at least $17$ prime factors), is not established by its proof: the proof combines an upper bound on $m$ with a lower bound on $n$ to get $p_1q_s<rs$, while the bounds $rn/p_r<m<rn/p_1$ and $sm/q_s<n<sm/q_1$ give only $p_1q_1<rs<p_rq_s$, as Bonfioli's manuscript shows (§14, p. 63; its claim page). The conclusion $r+s\ge34$ itself follows from Rosen's bound $|P\cup Q|\ge59$, recorded on the problem page.

Acceptance. Refereed: J. Kovič, The arithmetic derivative and antiderivative, J. Integer Seq. 15 (2012), Article 12.3.8; received 19 May 2011, revised 18 March 2012, published 25 March 2012 (the journal's article page). The site labels the problem VERIFIABLE and does not cite the paper, so no reviewed evidence is listed. This claim is partial, so the problem stays open.