Erdős Problem #378

FORMAL RESTATEMENT

Erdős #378; reconstruction with endpoint correction, 2026-09-24. Let \[ S(n)=\#\{1\leq j\leq n-1:\ \binom nj\text{ is squarefree}\}. \] For each fixed positive integer $r$, establish existence of a positive natural density for $\{n:S(n)\geq r\}$.

QUICK LITERATURE AND CONTEXT CHECK

Granville and Ramaré proved this in Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Their Theorem 5 counts the whole Pascal row, including $\binom n0=\binom nn=1$. Thus their $2m+2$ squarefree entries correspond to $2m$ internal entries. Applying $2m+2$ directly to the problem's internal count would give an incorrect index shift.

OBJECTIVE AND ATTACK PLAN

Use the published distribution theorem with its exact endpoint convention. Prove the parity reduction, derive a finite complementary sum for each requested density, and give an elementary positive lower bound for the first two thresholds.

WORK

External inputs, stated precisely. Theorem 5 of Granville--Ramaré gives a natural density $\eta_m$ for the integers $n$ whose whole Pascal row has exactly $2m+2$ squarefree entries, for every integer $m\geq0$. Moreover $\eta_m>0$ for every $m\geq1$. Their Theorem 1 states that $\binom{2a}{a}$ is not squarefree for $a>4$. These two results are used as published inputs; the exponential-sum and sieve estimates establishing them are not reproduced here.

Why the number of internal entries is even. The involution $j\mapsto n-j$ preserves $\binom nj$ and pairs all internal indices except possibly $j=n/2$. For odd $n$ there is no fixed index. For even $n>8$ the fixed central coefficient is not squarefree by the second input. Hence $S(n)$ is even for every $n>8$.

Since the two endpoints always contribute two squarefree entries, the set \[ E_m=\{n:S(n)=2m\} \] has density $\eta_m$. For $n>8$, failure of $S(n)\geq r$ is exactly membership in one of the finitely many disjoint sets $E_m$ with $2m<r$. Therefore the desired density exists and is \[ \boxed{\ \delta_r=1-\sum_{m=0}^{\lceil r/2\rceil-1}\eta_m\ }. \tag{1} \] Only a finite union is involved in this deduction, so no interchange of an infinite sum and a density limit is being assumed. If $m\geq\max(1,\lceil r/2\rceil)$, then $E_m\subseteq\{n:S(n)\geq r\}$ and $\eta_m>0$; hence $\delta_r>0$. Also $1$ gives $\delta_{2h-1}=\delta_{2h}$ for $h\geq1$, reflecting the parity pairing.

An elementary check on the first thresholds. For squarefree $n\geq3$, the two distinct entries at $j=1,n-1$ both equal $n$, so $S(n)\geq2$. The identity \[ \mathbf1_{\text{squarefree}}(n)=\sum_{d^2\mid n}\mu(d) \] gives \[ \#\{n\leq x:n\text{ squarefree}\} =\sum_{d\leq\sqrt x}\mu(d)\lfloor x/d^2\rfloor =\frac{x}{\zeta(2)}+O(\sqrt x). \] Here the rounding errors total $O(\sqrt x)$ and the omitted absolutely convergent series tail contributes $O(\sqrt x)$. Thus \[ \delta_1=\delta_2\geq\frac6{\pi^2}. \] This independently verifies positivity at the first two thresholds, but does not identify their exact density.

RESULT AND LIMITATIONS

Attempt status: solved relative to the two explicitly stated Granville--Ramaré theorems. Formula $1$, positivity for every fixed $r$, and the endpoint and parity conventions are fully justified. The deep published distribution theorem remains an external dependency; no new distribution theorem or numerical values of $\eta_m$ are claimed.

SOURCES

https://www.erdosproblems.com/378; Granville and Ramaré, Theorems 1 and 5 (printed pages 1 and 3): https://dms.umontreal.ca/~andrew/PDF/ramare.pdf.

Completion rate estimate: 100% relative to the stated external inputs.