Browse Problems

Does there exists a constant `c` such that `f n - 2 * n ~ c * (n / log n)`?

Does there exists a constant `c` such that `f n - 2 * n ~ c * (n / log n)`?

`f n - 2 * n = θ (n / log n)`. This is proved in [EGS82].

`f n - 2 * n = θ (n / log n)`. This is proved in [EGS82].

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} …

Cambie has observed that a positive answer follows from the result above with $a_t \leq n$, simply by pairing variables together, e.g. taking $a'_i = a_{2i-1}a_{2i}$ (and the lower bound follows from …

If we change the condition to $a_t \leq n$ it can be shown that $$ A(n) = n - \frac{n}{\log n} + o\left(\frac{n}{\log n}\right) $$

They proved (with Selfridge) that this holds for $n=10$.

They ask about the behaviour of $t_{n-3}(n!)$ and also ask whether, for infinitely many $n$, $t_k(n!)< t_{k-1}(n!)-1$ for all $1\leq k < n$.

In [ErGr80] they mention a conjecture of Erdős that the sum is $o(x^2)$. This was proved by Erdős and Hall [ErHa78], who proved that in fact $\sum_{n\leq x}t_2(n)\ll \frac{\log\log\log x}{\log\log x}x^2.$

Erdős and Hall conjecture that the sum is $o(x^2/(\log x)^c)$ for any $c<\log 2$.

Since $t_2(p)=p-1$ for prime $p$ it is trivial that $\sum_{n\leq x}t_2(n)\gg \frac{x^2}{\log x}$.

Is it true that $\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}$ for some $c>0$?

Is it true that, for $k\geq 2$, $\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?$

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 $c=8a^2+8a+1$, …

**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 …

Cambie observed that there are no solutions to `n! = x^4 + y^4` with `x.Coprime y` and `x*y > 1`.

Erdős and Obláth proved there are no solutions when `x.Coprime y` and `k ≠ 4`.

Pollack and Shapiro proved there are no solutions to `n! = x^4 - 1`.

The only solution to `n! = x^2 + y^2` with `x*y > 1` is `6! = 12^2 + 24^2` (up to swapping `x` and `y`).

Is it true that, for any $C > 0$, there infinitely many $n$ such that: $$ p_{n + 1} - p_n > C \frac{\log\log n\log\log\log\log n}{(\log\log\log n) ^ 2}\log n $$

For what functions $g(N) → \infty$ is it true that $$\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$?

Prove that, for any finite set $A\subset\mathbb{N}$, there exist $a, b\in A$ such that $$ \gcd(a, b)\leq a/|A|. $$

A conjecture of Graham [Gr70], who also conjectured that (assuming $A$ itself has no common divisor) the only cases where equality is achieved are when $A = \{1, \dots, n\}$ or $A …

Proved for all sufficiently large sets (including the sharper version which characterises the case of equality) independently by Szegedy [Sz86] and Zaharescu [Za87]. The following is taken from [Sz86]. There exists an …

Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$?

If we only allow the digits $1$ and $2$ then $2^{15}$ seems to be the largest such power of $2$.

Can infinitely many $n$ reach the same prime under the iteration $n\mapsto\phi(n) + 1$?

What is the density of $n$ which reach any fixed prime under the iteration $n\mapsto\phi(n) + 1$?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = O(g(n))$?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = o(g(n))$?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached. What is $\Theta(c(n))$?

Is it true that iterates of $n\mapsto\sigma(n) - 1$ always reach a prime?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = O(g(n))$?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = o(g(n))$?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. What is $\Theta(c(n))$?

Let $σ_1(n) = σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$. Is it true that $\lim_{k → ∞} σ_k(n)^{\frac 1 k} = ∞$? This is problem (iii) from Erdos, Granville, …

Let $σ_1(n)=σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$. Is it true that, for every $m, n ≥ 2$, there exist some $i, j$ such that $σ_i(m) = σ_j(n)$?

Are there infinitely many barriers for `ω`?

Erdős believed there should be infinitely many barriers for `Ω`, the total prime multiplicity.

Selfridge computed that the largest `Ω`-barrier below `10^5` is `99840`.

Does there exist some `ε > 0` such that there are infinitely many `ε`-barriers for `ω`?

Erdős proved that the barrier set for `expProd` is infinite and even has positive density.

Let $h_1(n) = h(n)$ and $h_k(n) = h(h_{k-1}(n))$. Is it true, for any $m,n$, there exist $i$ and $j$ such that $h_i(m) = h_j(n)$?

Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable. Does `V(2x)/V(x)→2` ?

Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable. Is there an asymptotic formula for `V(x)`?