Problem 700

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claims/: The 1 claim page of Problem 700, one per claimant's result; the problem's standing derives from them.

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Statement. Let

$$f(n)=\min_{1<k\leq n/2}\textrm{gcd}\left(n,\binom{n}{k}\right).$$

Characterise those composite $n$ such that $f(n)=n/P(n)$, where $P(n)$ is the largest prime dividing $n$. Are there infinitely many composite $n$ such that $f(n)>n^{1/2}$? Is it true that, for every composite $n$,

$$f(n) \ll_A \frac{n}{(\log n)^A}$$

for every $A>0$?

Formulation. The site, like the formal-conjectures file, takes $P(n)$ to be the largest prime dividing $n$. Erdős and Szekeres [ErSz78] use $P$ for a greatest prime factor on p. 97, but at their inequality (7) on p. 98, $f(n)\le n/P(n)$ for composite $n$, they define $P(n)$ as the greatest prime power dividing $n$, and on p. 99 they ask to characterize the composite $n$ with $f(n)=n/P(n)$; their deduction (9) of $f(n)<(1+o(1))n/\log n$ from (7), which the site's commentary repeats, needs that reading. The two readings agree for squarefree $n$ but not in general: $f(12)=3$ equals $12/4$ but not $12/3$, and under the site's reading every prime square is an equality case, since $f(p^2)=p$. This page's standing targets the site's wording; the first question is open under both readings.

Status. Open, the site's label. The site credits GPT 5.6 Sol Pro, prompted by Price, with a positive answer to the second question, infinitely many products of three primes with $f(n)\sim n^{2/3}$ (page edited 28 August 2026); see the Price claim page, a pending partial claim. The standing in the frontmatter derives from the claim pages.

Source. erdosproblems.com/700, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #700, https://www.erdosproblems.com/700.

References.

• [ErSz78] Erdős, P. and Szekeres, G., Some number theoretic problems on binomial coefficients. Austral. Math. Soc. Gaz. 5 (1978), 97-99. Library home: erdos_1978_number_theoretic_problems_binomial_coefficients.

Formalization. The formal-conjectures file FormalConjectures/ErdosProblems/700.lean, linked at its commit of 2026-09-19, states the three questions with sorry and defines $P(n)$ as the largest prime factor; it records no formal proof.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1978_number_theoretic_problems_binomial_coefficients
• erdos_1978_number_theoretic_problems_binomial_coefficients / inequality_6
• erdos_1978_number_theoretic_problems_binomial_coefficients / inequality_7
• erdos_1978_number_theoretic_problems_binomial_coefficients / inequality_8
• erdos_1978_number_theoretic_problems_binomial_coefficients / inequality_9