Li 2026 erdos problem 684 at density one

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corollary_1_2: For almost all positive integers n, the least k at which the part of n choose k made of primes at most k exceeds n^2 is (2/(1 - gamma) + o(1)) log n, that is 4.7305... log n; Problem 684's f(n) at density one, not at every n.

proposition_5_3: For fixed A, delta > 0, all but o(X) integers n in [X, 2X) satisfy |log u(n,k) - (1 - gamma)k| <= delta log X simultaneously for every integer k up to A log X, with a quantitative count of exceptions around the mean m(k).

theorem_1_1: For each fixed c > 0, the least k at which the part of n choose k made of primes at most k exceeds n^c is (c/(1 - gamma) + o(1)) log n for all n outside a set of natural density zero; a normal-order result, not a bound at every n.

theorem_1_3: For n uniform in [X, 2X) and k tending to infinity with k at most A log X, log u(n,k) minus its complete-residue mean, divided by the square root of V(k) ~ (2 - log(2 pi)) k log k, tends to a standard normal law.

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Eric Li, Erdős Problem 684 at Density One: Small-prime Parts of Binomial Coefficients and Gaussian Fluctuations. arXiv:2606.08216v1 (6 June 2026), doi:10.48550/arXiv.2606.08216. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2606.08216), every other right reserved. The acknowledgements (p. 18) record the use of OpenAI's ChatGPT in preparing the manuscript, the author taking full responsibility for the accuracy of its final contents.

Reading scope. The definitions, theorem statements, proof structure, labels and page locators below were checked against the page images of the v1 PDF, whose printed and physical page numbers coincide. This is claims checking and a proof map, not proof verification.

Small-prime part and carry representation

For $0\leq k\leq n$, Li defines

$$u(n,k):=\prod_{p\leq k}p^{\nu_p\binom nk},$$

the largest divisor of $\binom nk$ supported on primes at most $k$, and, for fixed $c>0$,

$$f_c(n):=\min\{0\leq k\leq n:u(n,k)>n^c\},$$

with $f_c(n)=\infty$ if the set is empty (Section 1, printed/physical pp. 1--2). Kummer's theorem is used in the exact residue form

$$\nu_p\binom nk =\sum_{a\geq1}\mathbf 1_{[n]_{p^a}<[k]_{p^a}}, \qquad U_k(n):=\log u(n,k) =\sum_{p\leq k}\sum_{a\geq1}(\log p) \mathbf 1_{[n]_{p^a}<[k]_{p^a}},$$

where $[x]_q$ is the least nonnegative residue modulo $q$ (equation (1.1), printed/physical p. 3; Lemma 2.2 and its proof, p. 4). Thus divisibility by a fixed prime is encoded by the occurrence of at least one carry level $p^a$.

Complete-residue averaging gives

$$m(k):=\sum_{p\leq k}\log p\sum_{a\geq1}\frac{[k]_{p^a}}{p^a} =k\sum_{p\leq k}\frac{\log p}{p-1}-\log k! =(1-\gamma)k+o(k),$$

as $k\to\infty$, and, for each fixed $A>0$, $\sup_{1\leq k\leq AL}|m(k)-(1-\gamma)k|=o_A(L)$ as $L\to\infty$, used with $L=\log X$ (Lemma 2.3, printed/physical p. 5). The cancellation between the two terms of order $k\log k$ produces the constant $1-\gamma$.

Concentration and first crossing

Proposition 5.3 (printed/physical pp. 10--11) proves that, for fixed $A>0$ and $\delta>0$,

$$\#\left\{X\leq n<2X: \sup_{1\leq k\leq A\log X} |U_k(n)-(1-\gamma)k|>\delta\log X\right\}=o_{A,\delta}(X).$$

The estimate is simultaneous in every integer $k$ in the logarithmic window, but only outside an exceptional set of $n$ of density zero. Lemma 3.1 (printed/physical pp. 6--7) supplies one part of that set: its proof discards every $n$ with $p^a\mid n-b$ for some prime $p\leq A\log X$, some $0\leq b\leq A\log X$ and some prime power $p^a\in(X^{1/10},2X]$. The other part, $O_{A,\delta}(X(\log\log X)^2/\log X)$ integers, comes from the fourth-moment bound of Lemmas 5.1 and 5.2 and Markov's inequality (p. 11).

Theorem 1.1 (printed/physical p. 2; proof in Section 6, pp. 11--12) consequently shows, for each fixed $c>0$,

$$f_c(n)=\left(\frac{c}{1-\gamma}+o(1)\right)\log n$$

for almost all positive integers $n$. Corollary 1.2 (p. 2) specializes this to the Erdős Problem 684 threshold:

$$f_2(n)=\left(\frac{2}{1-\gamma}+o(1)\right)\log n =(4.730544237\ldots+o(1))\log n$$

for almost all $n$. This is a normal-order result, not a worst-case bound.

Gaussian fluctuations

For fixed $A>0$, $n$ uniform on $\mathcal I_X=[X,2X)\cap\mathbb Z$ and an integer-valued $k=k(X)\to\infty$ with $k\leq A\log X$, set

$$\alpha_p(k)=\frac{[k]_p}{p},\qquad V(k)=\sum_{p\leq k}(\log p)^2\alpha_p(k)(1-\alpha_p(k)).$$

Theorem 1.3 (printed/physical p. 3; proof on pp. 17--18) proves

$$V(k)=(2-\log(2\pi)+o(1))k\log k, \qquad \frac{U_k(n)-m(k)}{\sqrt{V(k)}}\Rightarrow\mathcal N(0,1),$$

together with $\mathbb E_XU_k(n)=m(k)+o(\sqrt{V(k)})$, $\operatorname{Var}_X(U_k(n))\sim V(k)$, and the corresponding fully standardized central limit theorem. The variance asymptotic is Lemma 7.1 (pp. 13--14), the prime-level central limit theorem is Lemma 7.2 (pp. 14--15), and the $L^2$-negligibility of the centered higher-prime-power contribution is Lemma 7.3 (pp. 15--17). Higher powers remain necessary in the mean; only after centering do the prime levels alone govern the Gaussian scale.

Relevance and limit for E0699

The carry formula is relevant to E0699 because it gives an exact local criterion for a prime to divide each binomial coefficient. For fixed $1\leq i<j\leq n/2$, a common prime $p$ would require a carry at at least one $p$-power level for each of $\binom ni$ and $\binom nj$. This makes Li's residue-indicator and Chinese-remainder framework potentially useful for studying the needed two-coefficient correlation.

The paper does not prove that correlation. Its variables sum the weighted valuations of one coefficient $\binom nk$ over primes $p\leq k$, whereas E0699 asks, for every triple $(n,i,j)$, for one same prime $p\geq i$ dividing both $\binom ni$ and $\binom nj$. Large values of the two small-prime parts could be supported on disjoint primes, and every prime counted for $u(n,i)$ is at most $i$, so only $p=i$ can meet E0699's lower cutoff $p\geq i$. Moreover, the concentration and Gaussian laws average over $n\in[X,2X)$ and permit an $o(X)$ exceptional set; E0699 is pointwise in every $n,i,j$, precisely where discarded congruence obstructions may matter. Density-one one-coefficient averages therefore do not establish a pointwise simultaneous-common-prime theorem.

Bears on. #699 (methodological context; does not resolve the pointwise common-prime question), #684 (Corollary 1.2, p. 2, gives the problem's $f(n)=f_2(n)$ as $(2/(1-\gamma)+o(1))\log n$ outside a set of natural density zero; no bound on $f(n)$ at an individual $n$, and the paper says it should not be quoted as a resolution of the pointwise problem, p. 18)

Results.

• Theorem 1.1 (p. 2): $f_c(n)=(c/(1-\gamma)+o(1))\log n$ for almost all $n$, each fixed $c>0$.
• Corollary 1.2 (p. 2): the case $c=2$, Problem 684's threshold.
• Theorem 1.3 (p. 3): Gaussian fluctuations of $U_k(n)$ for $k\to\infty$ with $k\le A\log X$.
• Proposition 5.3 (pp. 10--11): uniform concentration of $U_k(n)$ about $(1-\gamma)k$.

Source artifact. arXiv:2606.08216v1, submitted and dated 2026-06-06.

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