Browse Problems

Is there an absolute constant $K$ such that, for every $C > 0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + C n^{\frac{1}{4}})$.

Is there an absolute constant $K$ such that, for every $C > 0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + C n^{\frac{1}{4}})$.

A question of Erdős and Rosenfeld, who proved that there are infinitely many $n$ with $4$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + n^{\frac{1}{4}})$.

Erdős and Rosenfeld, ask whether $4$ is the best possible $K$ for the infinitude of $n$ with $K$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + n^{\frac{1}{4}})$.

What is the size of the largest subset `A` of `{1,...,n}` such that if `a ≤ b ≤ c ≤ d ∈ A` and `abcd` square then `ad=bc`

The primes show that `|A| ≫ n/log n` is possible.

`|A|=o(n)`.

Let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\leq i < k$. Is it true that $v_0(n)=\max_{k\geq 0}v(n,k)\to \infty$ as $n\to \infty$?

Let $v_l(n) = \max_{k\geq l} v(n,k)$. For every fixed $l$, $v_l(n) \to \infty$ as $n \to \infty$ [ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive …

$v_0(n) > 1$ for all $n$ except $n$ = 0, 1, 2, 3, 4, 7, 8, 16 [ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive …

Does $v_1(n) = 1$ have finite solutions? [ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.

Does $V_1(n) = 1$ have finite solutions? This is a modification of `erdos_889.variants.v1_eq_1_finite`, which might make it more amenable to attack according to [ErSe67]. [ErSe67] Erdős, P. and Selfridge, J. L., Some …

If $\omega(n)$ counts the number of distinct prime factors of $n$, then is it true that, for every $k\geq 1$, $\liminf_{n\to \infty}\sum_{0\leq i < k}\omega(n+i)\leq k+\pi(k)?$

Is it true that $\limsup_{n\to \infty}\left(\sum_{0\leq i < k}\omega(n+i)\right) \frac{\log\log n}{\log n}=1?$

A question of Erdős and Selfridge [ErSe67], who observe that $\liminf_{n\to \infty}\sum_{0\leq i < k}\omega(n+i)\geq k+\pi(k)-1$ for every $k$. This follows from Pólya's theorem that the set of $k$-smooth integers has unbounded …

It is a classical fact that $\limsup_{n\to \infty}\omega(n)\frac{\log\log n}{\log n}=1.$

Let $2=p_1 < p_2 < \cdots$ be the primes and $k\geq 2$. Is it true that, for all sufficiently large $n$, there must exist an integer in $[n,n+p_1\cdots p_k)$ with $>k$ many …

This is unknown even for $k=2$ - that is, is it true that in every interval of $6$ (sufficiently large) consecutive integers there must exist one with at least $3$ prime factors?

Schinzel deduced from Pólya's theorem [Po18] (that the sequence of $k$-smooth integers has unbounded gaps) that this is true with $p_1\cdots p_k$ replaced by $p_1\cdots p_{k-1}p_{k+1}$.

Weisenberg has observed that Dickson's conjecture implies the answer is no if we replace $p_1\cdots p_k$ with $p_1\cdots p_k-1$. Indeed, let $L_k$ be the lowest common multiple of all integers at most …

Does the limit $\lim_{n\to\infty} \frac{f(2n)}{f(n)}$ tend to infinity? (Other finite limits have been ruled out by [KoLu25], see below)

Kovač and Luca [KoLu25] (building on a heuristic independently found by Cambie (personal communication)) have shown that there is no finite limit, in that $\lim_{n\to\infty} \frac{f(2n)}{f(n)}$ is unbounded.

Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$ such that $\limsup_{p,k} f(p^k) \log(p^k) = ∞$. Is it true that $\limsup_n (f(n+1)−f(n))/ \log n = ∞$?

Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that $\limsup_{p,k} f(p^k) \log(p^k) = ∞$. Is it true that $\limsup_n f(n+1)/ f(n) = ∞$?

Wirsing [Wi70] proved that if $|f(n+1)−f(n)| ≤ C$ then $f(n) = c \log n + O(1)$ for some constant $c$. [Wi70] Wirsing, E., _A characterization of $\log n$ as an additive arithmetic …

Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that $\limsup_{p,k} f(p^k) \log(p^k) = ∞$ and $f(p^k) = f(p)$ or $f(p^k) = kf(p)$. Is it true that $\limsup_n (f(n+1)−f(n))/ …

Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that $\limsup_{p,k} f(p^k) \log(p^k) = ∞$ and $f(p^k) = f(p)$ or $f(p^k) = kf(p)$. Is it true that $\limsup_n f(n+1)/f(n) …

Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$. Is it true that $$ \limsup_{N\to\infty}\frac{|(A - A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} = \infty? $$ …

Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?

Does every set of $n$ distinct points in $\mathbb{R}^2$ contain at most $n^{1+O(\frac{1}{\log\log n})}$ many pairs which are distance $1$ apart?

Does there exists an entire non-zero transcendental function `f : ℂ → ℂ` such that for any sequence `n₀ < n₁ < ...`, `{ z | ∃ k, iteratedDeriv (n k) f …

Prove that there exists some $c>0$ such that $$h(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}$$ as $n\to \infty$.

Prove that there exists some $c>0$ such that $$h(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}$$ as $n\to \infty$.

Erdős and Selfridge prove in [Er82c] that $h(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}$.

A heuristic of Tao using the Cramér model for the primes suggests this is true with $c=\sqrt{2\pi}$.

Erdős and Selfridge prove in [Er82c] that $h(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}$.

A heuristic of Tao using the Cramér model for the primes suggests this is true with $c=\sqrt{2\pi}$.

Are there infinitely many $n$ such that if $$ n(n + 1) = \prod_i p_i^{k_i} $$ is the factorisation into distinct primes then all exponents $k_i$ are distinct?

If there are infinitely many primes $p$ such that $8p^2 - 1$ is prime, then this is true.

It is likely that there are infinitely many primes $p$ such that $8p^2 - 1$ is also prime.

Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$?

Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$?

Is it true that, for $k\geq 4$, $f_k(n) \gg \frac{n^{1-\frac{1}{k-1}}}{(\log n)^{c_k}}$ for some constant $c_k>0$?

It is known that $f_3(n)\asymp (n/\log n)^{1/2}$ (see [erdosproblems.com/1104]).

A positive answer to this question would follow from [erdosproblems.com/986]. The known bounds for that problem imply $f_k(n) \gg \frac{n^{1-\frac{2}{k+1}}}{(\log n)^{c_k}}.$

The lower bound $R(4,m) \gg m^3/(\log m)^4$ of Mattheus and Verstraete [MaVe23] (see [erdosproblems.com/166]) implies $f_4(n) \gg \frac{n^{2/3}}{(\log n)^{4/3}}$.

Graver and Yackel [GrYa68] proved that $f_k(n) \ll \left(n\frac{\log\log n}{\log n}\right)^{1-\frac{1}{k-1}}.$