Browse Problems

When $A =\{a_1 < \cdots\}$ corresponds to the set of primes, it is conjectured that the set of numbers $n$ that have representations \[n=\sum_{u\leq i\leq v}a_i\] has positive upper density.

Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. Show that …

Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. Show that …

Suppose monotone sequence $A$ satisfies the following: `A 0 = 1` and for all `j`, `A (j + 1)` is the smallest natural number that cannot be written as a sum of …

Find the value of the limit of `MinOverlapQuotient`!

Let $c > 0$ and $n$ be some large integer. What is the size of the largest set $A \subseteq \{1, \ldots, \lfloor c n \rfloor\}$ such that $n$ is not a …

Let $c > 0$ and $n$ be some large integer. What is the size of the largest set $A \subseteq \{1, \ldots, \lfloor c n \rfloor\}$ such that $n$ is not a …

Let $c > 0$ and $n$ be some large integer. What is the size of the largest set $A \subseteq \{1, \ldots, \lfloor c n \rfloor\}$ such that $n$ is not a …

There is no consecutive triple of powerful numbers.

Erdős [Er76d] conjectured a stronger statement: if $n_k$ is the $k$th powerful number, then $n_{k+2} - n_k > n_k^c$ for some constant $c > 0$. [Er76d] Erdős, P., Problems and results on …

Are there any $2$-full $n$ such that $n+1$ is $3$-full?

Are there infinitely many 3-full $n$ such that $n+1$ is 2-full?

The limit of `MinOverlapQuotient` exists and it is less than $0.385694$.

Find a better lower bound!

Find a better upper bound!

Are there infinitely many $n$ such that the largest prime factor of $n$ is $< n^{\frac{1}{2}}$ and the largest prime factor of $n + 1$ is $< (n + 1)^{\frac{1}{2}}$. Steinerberger has …

Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n+1) > P(n)$ has density $\frac{1}{2}$.

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.

Hickerson conjectured the largest solution the equation `n!=a_1!a_2!···a_k!`, with `n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k`, is `16!=14!5!2!`.

Show that if `P(n(n+1)) / log n → ∞` where `P(m)` denotes the largest prime factor of `m`, then the equation `n!=a_1!a_2!···a_k!`, with `n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k`, …

Show that if `P(n(n−1)) > 4 log n` for large enough `n`, where `P(m)` denotes the largest prime factor of `m`, then the equation `n!=a_1!a_2!···a_k!`, with `n−1 > a_1 ≥ a_2 ≥ …

Surányi was the first to conjecture that the only non-trivial solution to `a!b!=n!` is `6!7!=10!`.

Is `Erdos375Prop` true?

If `Erdos375Prop` is true, then `(n + 1).nth Prime - n.nth Prime < (n.nth Prime) ^ (1 / 2 - c)` for some `c > 0`.

In particular, if `Erdos375Prop` is true, then Legendre's conjecture is asymptotically true.

It is easy to see that for any `n ≥ 1` and `k ≤ 2`, if `n + 1, ..., n + k` are all composite, then there are distinct primes `p₁, …

There exists a constant `c > 0` such that for all `n`, if `k < c * (log n / (log (log n))) ^ 3 → (∀ i < k, ¬ (n …

Are there infinitely many $n$ such that ${2n\choose n}$ is coprime to $105$?

Erdős, Graham, Ruzsa, and Straus [EGRS75] have shown that, for any two odd primes $p$ and $q$, there are infinite many $n$ such that ${2n\choose n}$ is coprime to $pq$.

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 $n$?

Erdos, Graham, Ruzsa, and Straus proved that if $$ f(n) = \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} $$ and $$ \gamma_0 = \sum_{k = 2}^{\infty} \frac{\log k}{2^k} $$ then for almost …

Erdos, Graham, Ruzsa, and Straus proved that if $$ f(n) = \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} $$ and $$ \gamma_0 = \sum_{k = 2}^{\infty} \frac{\log k}{2^k} $$ then $$ \lim_{x\to\infty} …

Erdos, Graham, Ruzsa, and Straus proved that if $$ f(n) = \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} $$ and $$ \gamma_0 = \sum_{k = 2}^{\infty} \frac{\log k}{2^k} $$ then $$ \lim_{x\to\infty} …

Erdos, Graham, Ruzsa, and Straus proved that if $$ f(n) = \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} $$ then there is some constant $c < 1$ such that for all large …

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) …

Does there exist $B \subset \mathbb{N}$ which is not an additive basis, but is such that for every set $A \subseteq \mathbb{N}$ of Schnirelmann density $\alpha$ and every $N$ there exists $b …

Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of $$ \prod_{i = 0}^k (p ^ 2 + i) $$ is $p$?

Let $F(n) := \max\{m + p(m) \mid \textrm{$m < n$ composite}\}\}$ where $p(m)$ is the least prime divisor of $m$. Is it true that $F(n)>n$ for all sufficiently large $n$?

Let $F(n) := \max\{m + p(m) \mid \textrm{$m < n$ composite}\}\}$ where $p(m)$ is the least prime divisor of $m$. Does $F(n) - n \to \infty$ as $n\to\infty$?

A question of Erdős, Eggleton, and Selfridge, who write that in fact it is possible that this quantity is always at least $n+(1-o(1))\sqrt{n}$

There is a $k$, such that $2 \le k \le n - 2$ and $\binom{n}{k}$ can be the product of consecutive primes infinitely often?

For all $2 \le k \le n - 2$, can $\binom{n}{k}$ be the product of consecutive primes infinitely often?

Can $\binom{n}{2}$ be the product of consecutive primes infinitely often?

Is there an absolute constant `c > 0` such that, for all `1 ≤ k < n`, the binomial coefficient `n.choose k` has a divisor in `(cn, n]`?

It is easy to see that `n.choose k` has a divisor in `[n / k, n]`.

The following is Schinzel's conjecture, which appears in [Gu04].

Is it true for any `c < 1` and all `n` sufficiently large, for all `1 ≤ k < n`, `n.choose k` has a divisor in `(cn, n]`? This is a variant …

Is it true that for every $n \geq 1$ there is a $k$ such that $$ n(n + 1) \cdots (n + k - 1) \mid (n + k) \cdots (n + …

Is there an infinite Sidon set $A\subset \mathbb{N}$ such that $\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$ for all $\varepsilon > 0$?