Erdős Problem #414 (n + τ(n)): structural lemmas, a bound on the number of trajectories, and computations to 10^16
Year: 2026
Type: preprint
Abstract: Let h(n) = n + τ(n), where τ(n) is the number of divisors of n. Erdős and Graham (and, earlier, C. Spiro in 1977) asked whether the h-orbits of any two starting points eventually meet — Erdős Problem #414. This note collects elementary structural observations and computations on the problem. (i) Every starting point below 10^16 lies in a single component: the orbits of the 38 "frontier" integers at N = 10^16 (the values m + τ(m) with m < N ≤ m + τ(m), through which every orbit from below N must pass) all merge within N + 3.8·10^9; the same method reproduces exactly the verification to 10^10 of the Erdős Problem a Day report (White–Claude, July 2026), and an independent full forward pass confirms merging for all n ≤ 10^9. (ii) The residue class n ≡ 2 (mod 4) is a trap: a trajectory there cannot change parity until it lands exactly on some 2m² with m odd, and the trapped dynamics is conjugate to the map g(m) = m + τ(m) on odd integers (h(2m) = 2g(m)); the trajectory of 1 spends 97.8 % of its first 560,000 steps in this class. This reduces the parity question (necessary, not sufficient, for merging across parities) to whether every g-orbit meets infinitely many odd squares. (iii) Unconditionally at most log Y + 2γ + o(1) distinct trajectories can be alive at height Y — a bound that appears, with a sharper error term, in E. Li's preprint arXiv:2606.17926 (June 2026), whose "lower-runner races" also cover the braid structure described here; we add a refinement by residue class. (iv) Scans for further congruence obstructions (all moduli ≤ 36), for invariants of the pair (position, gap) modulo 8, 16, 24, 32, and for obstructions to collisions find only the mod-4 structure; hitting rates of odd squares and merge hazards match a random model with step size ≍ log n; adjacent pairs (n, n+1), to which the conjecture reduces, almost never merge quickly precisely because of the parity mechanism. Code (C, segmented divisor sieve) is included. This work was AI-assisted (Claude, Anthropic); all statements were checked by the author, and no proof of the conjecture is claimed.
Author: Szczupak, Elie
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212583273
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