Claims

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2026_07_22_ankit: Ankit's notes of 22 July 2026, linked from the problem's forum thread, bound the number of iterations of $n\mapsto\phi(n)+1$ by $5\sqrt n+1$, report every basin of a prime below $30000$ computed exactly and finite, and give a $2$-adic argument that all but $O(\sqrt x)$ primes $p\le x$ with $p\equiv3\pmod4$ have the basin $\{p,2p\}$.

2026_07_22_van_doorn: Wouter van Doorn's forum comment of 22 July 2026 derives $F(n)=O(\sqrt n)$ for the number $F(n)$ of iterations of $n\mapsto\phi(n)+1$ needed to reach a prime, from $\phi(n)\le n-\sqrt n$ for composite $n$. It improves the trivial bound $o(n)$; the claim is pending.

2026_10_19_plasma_ai_i: Plasma AI's tier-0 deduction from a 2005 exceptional-prime theorem shows that almost every prime $p$ is reached under $n\mapsto\phi(n)+1$ only from $p$ and $2p$, so its basin is finite and has density zero; catalog acceptance is pending.

2026_10_19_plasma_ai_ii: Every $n\ge2$ reaches a prime within $4(\sqrt n-\sqrt2)$ iterations of $n\mapsto\phi(n)+1$, and the basins of the primes $2,3,5,7,11$ are finite and determined completely, so Guy's lists for $5$ and $7$ are exhaustive; author-recorded by Plasma AI at tier 0, catalog acceptance pending.

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