Problem 647

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claims/: The 1 claim page of Problem 647, one per claimant's result; the problem's standing derives from them.

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Statement. Let $\tau(n)$ count the number of divisors of $n$. Is there some $n>24$ such that

$$\max_{m<n}(m+\tau(m))\leq n+2?$$

Status. Verifiable (the site's label, VERIFIABLE; page last edited 07 April 2026).

Source. erdosproblems.com/647, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #647, https://www.erdosproblems.com/647.

References.

• [Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.
• [Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.
• [Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.

Formalization. Statement in formal-conjectures.

Current assessment

The site labels the problem verifiable: a positive answer is witnessed by one integer $n>24$ together with the finite check of $m+\tau(m)\leq n+2$ for every $m<n$, so a solution could be confirmed by computation, whereas a negative answer would need a proof. The problem is Erdős and Selfridge's. The site's remarks (page last edited 07 April 2026) record that the inequality holds for $n=24$; that $n+2$ cannot be lowered, since $\max(\tau(n-1)+n-1,\tau(n-2)+n-2)\geq n+2$ for every $n\ge7$; that Erdős [Er79] found it extremely doubtful that infinitely many such $n$ exist and suggested that $\max_{m<n}(\tau(m)+m-n)\to\infty$; that Erdős [Er79d] wrote that it seems certain, though hopeless with the methods of the time, that for every $k$ infinitely many $n$ satisfy $\max_{n-k<m<n}(m+\tau(m))\leq n+2$, a statement that follows from Schinzel's Hypothesis H; and that Erdős [Er92e] offered a prize for an example, the prize the site shows. Tao's comment on the thread (2025-10-02), repeated in the remarks, places the problem among its neighbors: since $\tau(m)$ behaves like $2^{\omega(m)}$, it is similar to, though slightly weaker than, the first part of Problem 679 and much stronger than Problems 413 and 248. The formal-conjectures file, at its revision of 2026-09-18 (pinned), states the question and Erdős's two variants as open and proves the case $n=24$ by decision as erdos_647.variants.twenty_four.

No result settles an instance of the question, so the folder holds no accepted or pending claim, and one claim is rejected: Agbanwa 2026, an AI-assisted Zenodo write-up (first version 2026-01-18) asserting that no $n>24$ exists, with a Lean file. Terence Tao's reply on the thread (2026-01-28) found its asymptotic step unproven and only assumed in the Lean, and its April revision has a gap of its own, as the claim page records.

The thread's partial results, none of which decides the question:

• Reductions. Sayan Dutta (2026-01-18) derived from the values at $m=n-1,n-2,\dots$ that any $n>84$ satisfying the inequality is a multiple of $2520$; Kenta Kitamura (2026-05-29) re-derived, within Scott Hughes's prime-chain families, Dutta's condition that $(n-3)/3=840N-1$ is prime. Scott Hughes (2026-05-27 to 2026-06-08; repository) refined the modular reduction to $n=2520N$ with $N$ in $41$ residue classes modulo $46189$, which his repository states is checked in Lean, and gave a prime-chain reduction to two explicit families, from which the Brun sieve bounds the number $|C(x)|$ of solutions up to $x$ by $x/(\log x)^7$ up to a constant; companion manuscripts described as submitted claim $|C(x)|\leq x\exp(-(\log\log x)^{2-o(1)})$. A density bound does not decide whether $C$ is empty.
• Searches without a proof certificate. OEIS A087280 records no solution in $(24,10^{10}]$; Patrik Idén's report (Zenodo, 2026-06-13, revised 2026-06-30 and 2026-07-02) extends this to $10^{12}$ (the minimum gap of $224$ that it reports, near $n=10^{11}$, is the least of the values its log prints at multiples of $10^{10}$, not a minimum over the range: the gap $\max_{m<n}(m+\tau(m))-n$ falls to $3$, first at $n=35$); Hughes's frontier certificate (2026-06-15) covers $n\leq6.15\times10^{17}$, and the thread (2026-09-10) credits Hughes and bentrd with a frontier near $9.17\times10^{18}$; veljjanoski's GPU search (repository) found no solution up to $10^{20}$ (2026-09-10) and then up to $10^{22}$ (2026-09-13).
• Kernel-checked exclusions. Ibrahim Mian's Lean development (thread, 2026-08-17; repository) proves from stored factorization witnesses that no solution lies in $(24,10^8]$; the preprint of Mian and Siddique, arXiv:2608.17880 (2026-08-18), extends the kernel-checked range to $10^9$; eerot's development (2026-10-04; repository) reaches $10^{13}$.

A finite exclusion, however checked, leaves the existence question open, and the thread records no further proof attempt. Beyond these results the mathematics of the problem is unassessed in this wiki.

Search scope (2026-10-07): the site's problem page and remarks, its discussion thread (18 comments) and its empty proof-claims tab, the community database entry, the formal-conjectures file, the Zenodo records and repositories linked from the thread, and the arXiv record of Mian and Siddique. MathSciNet and zbMATH were not searched and X was not used.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1979_unconventional_problems_number_theory_math_mag