Unbounded logarithmic limsup in Erd\H{o}s Problem 684 via shifted carry scheduling

Abstract

For $1\leq k\leq n$, let $u(n,k)=\prod_{p\leq k}p^{\nu_p\binom nk}$ and $f(n)=\min\{1\leq k\leq n:u(n,k)>n^2\}$. The minimum is interpreted as $+\infty$ if the set is empty. Here $\nu_p(m)$ denotes the exponent of the prime $p$ in $m$. Erd\H{o}s Problem 684 asks for bounds on $f(n)$. We prove $\limsup_{n\to\infty} \frac{f(n)}{\log n}\frac{\log\log\log n}{\log\log n}\geq\frac12$. In particular, $\limsup_{n\to\infty}f(n)/\log n=\infty$, so no uniform estimate $f(n)=O(\log n)$ is possible. The proof constructs integers $n=tL_M-h-1$, where $L_M=\operatorname{lcm}(1,\ldots,M)$. Product-cell coding yields simultaneous two-sided approximations to $tL_M$ modulo every prime in $(M,K]$. Writing $n+1=tL_M-h$, the shift by $h$ folds both signs into the same one-sided carry region. The carries at levels at most $M$ are bounded by $\log\binom{h+k}{h}$. For the prime powers above $K$, a truncated CRT witness has modulus below the search range; exponential weighting then gives an exponentially small exceptional set. An elementary anchored-fibre lemma selects a multiplier satisfying both requirements. Together, these ingredients prove the stated bound unconditionally. The theorem has been formally verified in Lean 4 with Mathlib.