Is the $\{2,3\}$-part of $n(n+1)$ infinitely often much larger than $n\log n$? (Erdős #933)
Statement
Write $n(n+1)=2^k3^\ell m$ with $(m,6)=1$, so that $2^k3^\ell$ is the largest divisor of $n(n+1)$ composed only of the primes $2$ and $3$. Is it true that $$\limsup_{n\to\infty}\frac{2^k3^\ell}{n\log n}=\infty?$$
Acceptance. FULLY RESOLVES: a proof — machine-checkable preferred, else a full written proof — that $\limsup_n 2^k3^\ell/(n\log n)=+\infty$ (for instance an explicit infinite family of $n$ along which the ratio tends to infinity, with proof); OR a proof that the limsup is finite, i.e. an improvement of Mahler's $n^{o(1)}$ ceiling to a bounded constant. ADVANCES: exhibit an explicit infinite family of $n$ on which $2^k3^\ell/(n\log n)$ exceeds any prescribed constant strictly larger than the current best value $3/\log 2$, with proof; or prove $\limsup \ge C$ for a specific constant $C>3/\log 2$; or prove an unconditional upper bound on $2^k3^\ell$ strictly sharper than Mahler's $n^{1+o(1)}$ valid for all $n$. Deliver the construction/family with its correctness proof, or the improved bound with proof.
Background
Posed by Erdős [Er76d]. Mahler proved (as a special case of a more general theorem) that the $\{2,3\}$-part is small, $2^k3^\ell < n^{1+o(1)}$. In the other direction Erdős remarked that 'it is easy to see' that $2^k3^\ell > n\log n$ for infinitely many $n$; Steinerberger supplied a clean proof by taking $n=2^{3^r}$, which forces $k=3^r$ and (by the lifting-the-exponent lemma applied to $2^{3^r}+1$) $\ell=r+1$, giving $2^k3^\ell = 2^{3^r}3^{r+1} = \tfrac{3}{\log 2}\,n\log n$. This shows the limsup is at least $3/\log 2 > 1$, but leaves entirely open whether it is unbounded. Formalised in Lean as part of the Google DeepMind Formal Conjectures project (formal-conjectures/933). Listed as open on erdosproblems.com/933 (fetched 2026-07-21, status 'open'); no Erdős prize is attached. (Compare Erdős #368 on the largest prime factor of $n(n+1)$, a neighbouring venue problem.) Attacker's tool: high-throughput computation of the $\{2,3\}$-part of $n(n+1)$ over large ranges to locate $n$ with anomalously large ratios and to model the extreme-value statistics guiding a construction, together with analytic control of the joint $2$-adic and $3$-adic valuations of $n$ and $n+1$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #933 (T. F. Bloom) | website |
| REF-02 | Lean formalisation of Erdős #933 (Google DeepMind Formal Conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.