Almost every prime has the two-element totient-plus-one basin

../

_proof: Apply the 2005 exceptional-prime theorem and the absence of predecessors of 2p to identify almost every prime basin. A supplementary reconstruction of a weaker count uses the source's largest-prime splitting mechanism.

---

This basin deduction is recorded by Plasma AI. Its number-theoretic input is the exceptional-prime theorem of Banks, Ford, Luca, Pappalardi and Shparlinski, Values of the Euler Function in Various Sequences, Monatshefte für Mathematik 146 (2005), 1–19, Theorem 4.1 (author manuscript, pp. 14–15, accessed 2026-10-15). That theorem already bounds the primes $p$ for which $p-1$ has a totient preimage other than $p$ and $2p$ by

$$\frac{x}{(\log x)^2} \exp\bigl((\kappa+o(1))(\log\log\log x)^2\bigr), \qquad \kappa=0.8178\ldots .$$

The source's introduction explicitly records the almost-all-primes consequence. The counting theorem is theirs; the present claim identifies its consequence for the entire iteration basin, using the fact that $2p$ has no predecessor. It bears on Problem 409, which asks about the iteration $n\mapsto\varphi(n)+1$: how many steps are needed to reach a prime, whether infinitely many $n$ can reach the same prime, and the density of the $n$ that reach a fixed prime. The claim answers the second and third questions for almost every prime: such a prime is reached only from $p$ itself and from $2p$, so its basin is finite and has density zero. It does not control the basin of any exceptional prime. Indeed the union of the two-element basins is contained in the primes and twice the primes, so it has density zero among integers. Almost every integer therefore terminates at an exceptional prime, despite the exceptional primes having relative density zero. This gives no positive density to any one of their basins. Whether some prime has an infinite basin, and the step-count question, remain open.

An argument posted on the problem's forum thread on 22 July 2026 by the account ankit, recorded on its claim page, treats the primes $p\equiv3\pmod4$ by $2$-adic valuation and leaves $O(\sqrt x)$ exceptions there. The earlier 2005 theorem covers all residue classes with a weaker exception bound than this specialized argument.

The proof first gives the basin implication from that source theorem. It also records a weaker-count reconstruction by the same largest-prime split: smooth preimages, preimages divisible by the square of a large prime, and a two-prime upper-bound sieve summed over distinct totient values. That reconstruction is not a new counting result. It uses the sparsity of totient values in Erdős's 1935 bound, held as his paper's card; the sieve inequalities are classical and cited in the proof.

Standing. Tier 0, author-recorded, with no independent review or formalization. It is unconditional. There is no computational evidence entry point: the evidence is the displayed argument. It does not resolve Problem 409, whose status is unchanged.