Browse Problems

Or even $f(n) < n^{c/\log\log n}$ for some constant $c > 0$?

Is it true that $f(n)\leq n^{o(1)}$?

Is it true that, for every $r$, there is a $k$ such that if $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, then $$ \prod_{1\leq i\leq r}\prod_{m\in …

Erdos and Selfridge [ErSe75] proved that the product of consecutive integers is never a power (establishing the case $r=1$). Theorem 1 from [ErSe75]. It is implied from `erdos_930.variant.consecutive_strong`. [ErSe75] Erdős, P. and …

Let $k$, $l$, $n$ be integers such that $k \ge 3$, $l \ge 2$ and $n + k \ge p^{(k)}$, where $p^{(k)}$ is the least prime satisfying $p^{(k)} \ge k$. Then there …

Let $k_1 \geq k_2 \geq 3$. Are there only finitely many $n_2\geq n_1 + k_1$ such that $$ \prod_{1\leq i\leq k_1}(n_1 + i)\ \text{and}\ \prod_{1\leq j\leq k_2} (n_2 + j) $$ have …

Erdős thought perhaps if the two products have the same factors then $n_2 > 2(n_1 + k_1)$. It is an open question whether this is true when allowing a finite number of …

In fact there exist counterexamples, like this one found by AlphaProof.

Erdős was unable to prove that if the two products have the same factors then there must exist a prime between $n_1$ and $n_2$.

Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two integers $p_r < n < p_{r+1}$ all of whose prime factors are $< p_{r + 1} - …

Erdős could show that the density of $r$ such that at least one such $n$ exists is $0$.

Is $n! + 1$ powerful for finitely many $n$?

Is $n! - 1$ powerful for finitely many $n$?

Is $2^n + 1$ powerful for finitely many $n$?

Is $2^n - 1$ powerful for finitely many $n$?

Let $A=\{n_1 < n_2 < \cdots\}$ be the sequence of powerful numbers (if $p\mid n$ then $p^2\mid n$). Are there only finitely many three-term progressions of consecutive terms $n_k,n_{k+1},n_{k+2}$?

If $r≥4$ then can the sum of $r-2$ coprime $r$-powerful numbers ever be itself $r$-powerful?

Cambie has also found solutions when $r=8$.

Euler had conjectured that the sum of $k - 1$ many $k$-th powers is never a $k$-th power, but this is false for $k=5$, as Lander and Parkin [LaPa67] found $$27^5+84^5+110^5+133^5=144^5$$. [LaPa67] …

Cambie has found several examples of the sum of $r - 2$ coprime $r$-powerful numbers being itself $r$-powerful. For example when $r=5$ we have $$3761^5=2^8\cdot3^{10}\cdot 5^7 + 2^{12}\cdot 23^6 + 11^5\cdot 13^5$$.

If $r≥4$ are there infinitely many sums of $r-2$ coprime $r$-powerful numbers that are themselves $r$-powerful?

Cambie has also found solutions when $r=7$.

Are there infinitely many triples of coprime $3$-powerful numbers $a, b, c$ such that $a + b = c$?

Let $r \ge 3$. Is it true that the set of integers which are the sum of at most $r$ $r$-powerful numbers has density $0$?

It is not known if all large integers are the sum of at most $r$-many $r$-powerful numbers.

Is it true that the set of integers which are the sum of at most three cubes has density $0$?

Heath-Brown [He88] has proved that all large numbers are the sum of at most three $2$-powerful numbers. [He88] Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. (1988), 137--163.

The set of integers which are the sum of at most two $2$-powerful numbers has density $0$.

Is there some constant $c > 0$ such that $h(n) < (\log n)^{c + o(1)}$ and, for infinitely many $n$, $h(n) > (\log n)^{c - o(1)}$.

Let $A$ be the set of powerful numbers. Is is true that $1_A\ast 1_A(n)=n^{o(1)}$ for every $n$?

Let $k \ge 4$ and $r\ge 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?

The case $k=4$ and $r=1$ remains open: Are there $4$-critical graphs without any critical edges?

Let $k \ge 4$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>1$? This was conjectured …

Dirac's conjecture was proved, for $k=5$: There exists a graph $G$ with chromatic number $5$, such that every vertex is critical, yet every critical set of edges has size $>1$, or in …

Jensen [Je02] gave an construction for $k$-critical graphs without any critical edges for all $k ≥ 5$. [Je02] Jensen, Tommy R., Dense critical and vertex-critical graphs. Discrete Math. (2002), 63--84.

Lattanzio [La02] proved there exist $k$-critical graphs without critical edges for all $k$ such that $k - 1$ is not prime. [La02] Lattanzio, John J., A note on a conjecture of {D}irac. …

Martinsson and Steiner [MaSt25] proved for every $r \ge 1$ if $k$ is sufficiently large, depending on $r$, there exist a graph $G$ with chromatic number $k$ such that every vertex is …

Is it true that $F(x) \leq (\log x)^{O(1)}$?

Is there a constant $C > 0$ such that, for all large $x$, every interval $[x, x+(\log x)C]$ contains two integers with the same number of divisors?

Erdős and Mirsky [ErMi52] proved that $\frac{(\log x)^{1/2}}{\log\log x}\ll F(x)$.

Erdős and Mirsky [ErMi52] proved that $\log F(x) \ll \frac{(\log x)^{1/2}}$.

There are infinitely many $n$ such that $τ(n) = τ(n+1)$. Proved in [He84]. Here τ is the divisor counting function, which is `σ 0` in mathlib.

The number of $n \le x$ with $τ(n) = τ(n+1)$ is at least $x / (\log x)^7$ for all sufficiently large $x$. Proved in [He84].

Improved lower bound in [Hi85]: $Ω(x / (\log \log x)^3)$.

There are infinitely many $n$ such that $τ(n) = τ(n + 5040)$. Proved in [Sp81].

Upper bound in [EPS87]: $O(x / \sqrt{\log \log x})$.

Let $S \subseteq \mathbb{R}$ be a set containing no solutions to $a + b = c$. Must there be a set $A \subseteq \mathbb{R} \setminus S$ of cardinality continuum such that $A …

Let $S\sub \mathbb{R}$ be a Sidon set. Must there be a set $A\sub \mathbb{R}∖S$ of cardinality continuum such that $A + A \sub \mathbb{R}∖S$?

If `a` has property `Erdos951_prop`, is it true that `#{a i ≤ x} ≤ π x`?

Beurling conjectured that if the number of Beurling integer in `[1, x]` is `x + o(log x)`, then `a` must be the sequence of primes.