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Claim. For a positive integer $n$ let $F(n)$ be the number of iterations of $n\mapsto\phi(n)+1$ needed before a prime is reached, the count that the erdosproblems.com remarks on Problem 409 call $F(n)$. Then $F(n)=O(\sqrt n)$. The comment derives this from the inequality $\phi(n)\le n-\sqrt n$ for composite $n$. It proves neither the inequality nor the deduction, and it gives no constant.

Submission note. Posted to the erdosproblems.com forum by Wouter van Doorn (account Woett) on 22 July 2026, in the problem's thread:

Small observation: from the inequality $\varphi(n) \le n - \sqrt{n}$ for composite $n$, it follows that $F(n) = O(\sqrt{n})$.

Covers. An upper bound for the first question of Problem 409, how many iterations are needed. erdosproblems.com reads that question as asking for good upper bounds on $F(n)$, and records Cambie's remark that $F(n)=o(n)$ is trivial. That trivial bound is the only upper bound the problem page lists from before 22 July 2026, and $O(\sqrt n)$ improves it. The claim gives no lower bound and does not determine the order of $F(n)$. The second and third questions, on the starting values that reach a fixed prime, are not addressed.

Claimant. Wouter van Doorn, whose erdosproblems.com profile gives that name for the account Woett, posted the comment in the problem's thread. The comment discloses no AI assistance. erdosproblems.com has not marked it as addressed by an update of the problem page.

Later explicit bounds. Two later write-ups give the bound with a constant, from the same inequality. The notes of the account ankit (their claim page), uploaded to GitHub on 22 July 2026 and linked from that account's comment posted later the same day in the same thread, state $F(n)\le5\sqrt n+1$ for all $n\ge1$ (their Theorem 1(vii)), proved by counting the composite iterates in dyadic ranges. They list it as apparently new and do not mention this comment. Plasma AI's research page on the small prime basins proves the inequality from the least prime factor $p\le\sqrt n$ of a composite $n$. It then shows that each composite step lowers $\sqrt n$ by at least $1/4$, so a start $n\ge2$ takes at most $4(\sqrt n-\sqrt2)$ nonfixed steps. That is Plasma AI's later result, author-recorded and without independent review, and pending on its claim page. Plasma AI's other claim page, Plasma AI's two-element basins, concerns the second and third questions and is independent of this bound.

Depends on. No page of this wiki.

Standing. Claimed. The claim is a one-line forum comment that states the inequality and the conclusion without proof. It has no refereed publication and no Lean proof. The erdosproblems.com page, last edited 20 December 2025, does not mention it, and its proof-claims tab is empty (accessed 2026-10-18). Plasma AI's research page above proves the inequality and an explicit form of the bound, author-recorded and without independent review; that is coverage, not acceptance.