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908 of 1221 statements are formalized in Lean (Formal Conjectures) · 107 carry an Erdős prize · 3 are independent of ZFC and 9 more can't be proved, or can't be disproved, from ZFC alone · “open” counts the computable and Lean-checked ones too.
Show that, for any \(n\geq 5\), the binomial coefficient \(\binom{2n}{n}\) is not squarefree.
Are there infinitely many \(n\) such that \({2n\choose n}\) is coprime to \(105\)?
Is there some absolute constant \(C > 0\) such that \[ \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} \leq C \] for all \(n\)?
Let \(S(n)\) denote the largest integer such that, for all \(1 ≤ k < n\), the binomial coefficient \(\binom{n}{k}\) is divisible by \(p^S(n)\) for some prime \(p\) (depending on \(k\)).Then \(\limsup S(n) = \infty\). This was formalized in…
If \(1<k<n-1\) then \(\binom{n}{k}\) is divisible by a prime \(p\leq n/2\) (except \(\binom{7}{3}=\binom{7}{4}=5\cdot 7\)). A conjecture of Erdős and Selfridge [ErGr80]. Proved by Ecklund [Ec69], who made the stronger conjecture that…
Let \(2 \le k \le n - 2\). Can \(\binom{n}{k}\) be the product of consecutive primes infinitely often? Here \(k\) may vary with \(n\): the question asks for infinitely many admissible binomial coefficients, not for a single \(k\) that…
Is there an absolute constant \(c > 0\) such that, for all \(1 \leq k < n\), the binomial coefficient \(\binom{n}{k}\) has a divisor in \((cn, n]\)? Bui, Naprienko, Pratt, and Zaharescu [BNPZ26] answered this negatively.
Is it true that for every \(k\) there exists \(n\) such that \[\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?\]
Are there only finitely many solutions to \[ \prod_i \binom{2m_i}{m_i}=\prod_j \binom{2n_j}{n_j} \] with the \(m_i,n_j\) distinct? Somani, using ChatGPT, has given a negative answer. In fact, for any \(a\geq 2\), if \(c=8a^2+8a+1\),…
Let \(P(n, k)\) be the largest prime factor of \(\binom{n}{k}\). There exists \(c > 0\) such that \(P(n, k) \ge \min(n - k + 1, k^{1 + c})\) for all \(0 < k \le n/2\). Erdős stated this for \(1 \le k \le n\) with the bound…
Let \(f(n)\) be the smallest \(0 \leq k \leq n\) for which the factor of \(\binom{n}{k}\) supported on primes at most \(k\) exceeds \(n^2\). Give bounds for \(f(n)\). An internal OpenAI model [APSSV26] has given an elementary argument…
Let \(\epsilon > 0\) and let \(n\) be large depending on \(\epsilon\). Is it true that for all \(n^{\epsilon} < k \leq n^{1-\epsilon}\) the number of distinct prime divisors of \(\binom{n}{k}\) is \[(1+o(1))\, k \sum_{k < p < n}…
Is there some \(h(n)\to \infty\) such that for all \(2\leq i<j\leq n/2\) \[\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \geq h(n)?\] This was resolved by Bergman [Be11], who proved that for any \(2\leq i<j\leq n/2\)…
Sylvester and Schur: for \(1 \le i \le n/2\) there is a prime \(p > i\) dividing n.choose i.
Let \(f(n) = \min_{1 < k \le n/2} \gcd(n, \binom{n}{k})\) and let \(P(n)\) be the largest prime dividing \(n\). **(a)** Characterise those composite \(n\) such that \(f(n) = n/P(n)\). Erdős–Szekeres [ErSz78] note that \(f(n) = n/P(n)\)…
Are there infinitely many pairs of integers \(n < m\) such that \(\binom{2n}{n}\) and \(\binom{2m}{m}\) have the same set of prime divisors? Yes: there are infinitely many consecutive pairs \((n, n+1)\). The formal proof registered as…
Find some reasonable function \(f(n)\) such that, for almost all integers \(n\), the least integer \(m\) such that \(m\nmid \binom{2n}{n}\) satisfies \[m\sim f(n).\] Here "almost all" means natural density \(1\), and \(m \sim f(n)\) means…
Singmaster's conjecture
Is it true that, for every integer \(t\geq1\), there is some integer \(a\) such that \({n \choose k} = a\) with \(1\leq k \le \frac{n}{2}\) has exactly \(t\) solutions?
Are there infinitely many binomial coefficients with deficiency 1?
For all \(n\ge 2k\) the least prime factor of \(\binom{n}{k}\) is \(\le\max(n/k,k)\), with only finitely many exceptions.