Is the sum of 1/p over primes p ≤ n not dividing $\binom{2n}{n}$ bounded uniformly in n? (Erdős #377)
Statement
Is there some absolute constant $C>0$ such that $$\sum_{p\leq n}1_{p\nmid \binom{2n}{n}}\frac{1}{p}\leq C$$ for all $n$, where the summation is restricted to primes $p\leq n$ and $1_{p\nmid \binom{2n}{n}}$ is the indicator that $p$ does not divide the central binomial coefficient $\binom{2n}{n}$? By Kummer's theorem, $p\nmid\binom{2n}{n}$ exactly when adding $n+n$ in base $p$ produces no carries, i.e. when every base-$p$ digit of $n$ is at most $(p-1)/2$.
Acceptance. FULLY RESOLVES: a proof that $\sup_n f(n)<\infty$ (an explicit admissible constant $C$ is a plus), OR a proof that $f(n)$ is unbounded, exhibiting the growth along an explicit sequence of $n$. Machine-checkable proof (Lean/Coq) preferred; otherwise a complete written proof with all steps. A finite computation alone cannot settle boundedness. ADVANCES: (a) improve the upper bound $f(n)\leq c\log\log n$ (the $c<1$ result stated in the background) to a strictly smaller order of growth, e.g. $f(n)=o(\log\log n)$ or $f(n)\ll(\log\log n)^{1-\delta}$, with proof; (b) prove the implied claim $\sum_{p\leq n}1_{p\mid\binom{2n}{n}}\frac{1}{p}=(1-o(1))\log\log n$; (c) a large-scale reproducible computation of $f(n)$ for all $n\leq N$ for an explicitly stated $N$, reporting the running maxima, the extremal $n$, and code — establishing the first published record table for this quantity; or (d) a Lean formalisation of the EGRS mean-value theorem. Deliver the proof file, or the computation code plus record tables.
Background
A question of Erdős, Graham, Ruzsa, and Straus [EGRS75], repeated in [Er79] and Erdős–Graham [ErGr80]; listed as open on erdosproblems.com/377 (fetched 2026-07-13, status 'open') and mentioned in problem B33 of Guy's collection [Gu04]. Writing $f(n)$ for the sum in question, EGRS75 proved the mean-value results $\lim_{x\to\infty}\frac{1}{x}\sum_{n\leq x}f(n)=\sum_{k=2}^{\infty}\frac{\log k}{2^k}=\gamma_0$ and $\lim_{x\to\infty}\frac{1}{x}\sum_{n\leq x}f(n)^2=\gamma_0^2$, so $f(n)=\gamma_0+o(1)$ for almost all $n$; the question is whether the exceptional $n$ can push $f$ to infinity. They also proved $f(n)\leq c\log\log n$ for all large $n$ with some constant $c<1$, while the trivial Mertens bound gives $(1+o(1))\log\log n$. A positive answer would imply $\sum_{p\leq n}1_{p\mid\binom{2n}{n}}\frac{1}{p}=(1-o(1))\log\log n$, of which EGRS say there is 'no doubt' it is true. The statement is formalised in Lean in the google-deepmind/formal-conjectures repository. Closely related to the venue problem on whether infinitely many $\binom{2n}{n}$ are coprime to $105$ (Erdős #376) — both concern which primes fail to divide the central binomial coefficient — but the present question is about the aggregate reciprocal sum over all primes, not three fixed primes. The attacker's tool: fast evaluation of $f(n)$ via the base-$p$ digit criterion to chart record values and the structure of extremal $n$ over large ranges; a proof needs simultaneous control of digit conditions across many primes.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #377 (T. F. Bloom) | website |
| REF-02 | Formalised statement of Erdős #377 (google-deepmind/formal-conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.