Do all iterated-$\sigma$ orbits eventually merge: $\sigma_i(m)=\sigma_j(n)$ for some $i,j$? (Erdős #412)
Statement
Let $\sigma$ be the sum-of-divisors function and $\sigma_k$ its $k$-fold iterate: $\sigma_1(n)=\sigma(n)$ and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that for every $m,n\ge2$ there exist $i,j\ge1$ with $$\sigma_i(m)=\sigma_j(n)\,?$$ In words: do the forward orbits of any two starting values under iterated $\sigma$ always eventually intersect, so that there is essentially a single trajectory on which all orbits coalesce?
Acceptance. FULLY RESOLVES: a complete rigorous proof that for all $m,n\ge2$ the orbits merge (some $\sigma_i(m)=\sigma_j(n)$), OR a rigorous disproof — a specific pair $m,n$ together with a PROOF that their orbits never coincide (note: merely checking many terms is not a proof of non-merging); full written proof. ADVANCES (each itself checkable): a proof that orbits merge for an explicit infinite family of pairs $(m,n)$; a proven structural obstruction showing that certain pairs cannot merge; a rigorous criterion deciding merging for a restricted class of pairs; or a reproducible large-scale computation that, for many pairs, either finds the merging indices $(i,j)$ or certifies non-merging up to a stated bound (code + range + the pairs and outcomes), materially extending Selfridge's numerical evidence. Deliver the proof, the proven witness pair, or the computation code with its certified range and tabulated merge / non-merge outcomes.
Background
Erdős [Er79d] attributes this conjecture to van Wijngaarden, who told it to him in the 1950s; it also appears in Erdős–Graham [ErGr80]; listed as open on erdosproblems.com/412 (fetched 2026-07-13, status 'open', tagged 'number theory | iterated functions'), no prize. Related OEIS sequences are A007497 (the orbit of $2$ under $\sigma$) and A051572. A formal statement exists in DeepMind's formal-conjectures Lean library. The conjecture asserts that the iterated-$\sigma$ dynamics eventually settles onto a single 'universal' trajectory that every orbit joins. The evidence points the other way: Selfridge reported numerical data suggesting the answer is NO (orbits that appear never to merge), and Erdős and Graham wrote that 'it seems unlikely that anything can be proved about this in the near future'. The erdosproblems.com page cross-references two neighbouring problems, Erdős #413 (erdosproblems.com/413) and #414 (erdosproblems.com/414). Attacker's tool: large-scale computation of $\sigma$-orbits to test pairwise merging (does the orbit of $m$ ever hit the orbit of $n$?), quantifying how far apart orbits stay and extending the evidence Selfridge began; a rigorous resolution would need analytic control of the multiplicative structure of $\sigma_k(n)$, which currently seems out of reach.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #412 (T. F. Bloom) | website |
| REF-02 | OEIS A007497 — orbit of 2 under the sum-of-divisors function sigma | website |
| REF-03 | OEIS A051572 — sequence related to iterating the sum-of-divisors function | website |
| REF-04 | DeepMind formal-conjectures — Lean statement of Erdős #412 | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.