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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.
practical numbers
**Conjecture 1.** Are there infinitely many practical numbers \(m\) such that \(h(m) < (\log \log m)^{O(1)}\)? More precisely: does there exist a constant \(C > 0\) such that for infinitely many practical numbers \(m\), we have…
Show that the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only finitely many solutions.
Does there exists a constant c such that f n - 2 * n ~ c * (n / log n)?
Let \(t(n)\) be maximal such that there is a representation \[n!=a_1\cdots a_n\] with \(t(n)=a_1\leq \cdots \leq a_n\). Obtain good bounds for \(t(n)/n\). In particular, is it true that \[\lim \frac{t(n)}{n}=\frac{1}{e}?\] It is easy to…
Let \(A(n)\) denote the least value of \(t\) such that \[ n! = a_1 \cdots a_t \] with \(a_1 \leq \cdots \leq a_t\leq n^2\). Then \[ A(n) = \frac{n}{2} - \frac{n}{2\log n} + o\left(\frac{n}{\log n}\right). \]
Let \(f(n)\) denote the minimal \(m\geq 1\) such that \[n! = a_1\cdots a_t\] with \(a_1<\cdots <a_t=a_1+m\). What is the behaviour of \(f(n)\)? We formalise the question as: does \(f(n) \to \infty\)? A result of Luca [Lu02] implies a…
Brocard-Ramanujan conjecture
**Brocard's Problem** Does \(n! + 1 = m^2\) have integer solutions other than \(n = 4, 5, 7\)?
Is it true that there are no solutions to n! = x^k ± y^k with x,y,n ∈ ℕ, x*y > 1, and k > 2? The answer is no: Jonas Barfield found the counterexample 10! = 48^4 - 36^4 (equivalently, 10! + 36^4 = 48^4). This is discussed in problem D2 of…
Can one show that \(\sum_{n\leq x}g_k(n) \sim c_k x\log x\) for some constant \(c_k\)?
Is there some function \(f(r)\) such that \(f(r)\to \infty\) as \(r\to\infty\), such that, for infinitely many \(n\), there exist \(a_1,a_2\) with \[a_1+a_2> n+f(r)\log n\] such that \(a_1!a_2! \mid n!2^n3^n\cdots p_r^n\)? It is ambiguous…
Does the equation \[2^m=a_1!+\cdots+a_k!\] with \(a_1<a_2<\cdots <a_k\) have only finitely many solutions? Asked by Burr and Erdős. Frankl and Lin [Li76] independently showed that the answer is yes, and the largest solution is…
Let \(p\) be an odd prime. Is it true that the equation \((p-1)!+a^{p-1}=p^k\) has only finitely many solutions? Originally proposed by Erdős and Graham [ErGr80]. Brindza and Erdős [BrEr91] proved that there are finitely many such…
If \(\tau(n)\) counts the number of divisors of \(n\), then what is the set of limit points of \[ \frac{\tau((n+1)!)}{\tau(n!)}? \] The limit points are exactly \(\{1\} \cup \{1+1/k : k \geq 1\}\).
Let \(p\) be a prime and \[A_p = \{ k! \pmod{p} : 1\leq k<p\}.\] Is it true that \[\lvert A_p\rvert \sim (1-\tfrac{1}{e})p?\]
Let \(p_1,\ldots,p_k\) be distinct primes. Are there infinitely many \(n\) such that \(n!\) is divisible by an even power of each of the \(p_i\)? The answer is yes, proved by Berend [Be97], who further proved that the sequence of such…
Let \(k ≥ 2\). Does \(((n+k)!)^2∣(2n)!\) hold for infinitely many \(n\)?
Let \(\varepsilon\) be sufficiently small and \(C, C' > 0\). Are there integers \(a, b, n\) such that \[a, b > \varepsilon n\quad a!\, b! \mid n!\, (a + b - n)!, \] and \[C \log n < a + b - n < C' \log n ?\] Note that the website currently…
Let \(C>0\) be a constant. Are there infinitely many integers \(a,b,n\) with \(a+b> n+C\log n\) such that the denominator of \[\frac{n!}{a!b!}\]contains only primes \(\ll_C 1\)? Erdős [Er68c] proved that if \(a!b!\mid n!\) then…
Prove that there exists some \(c>0\) such that \[h(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}\] as \(n\to \infty\).
For each \(k \geq 2\), does the set \(A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}_{\geq 1}\text{ finite}\right\}\) of all finite sums of distinct factorials contain only finitely many \(k\)-th powers?