Asks for the order of magnitude of the error term $E(x)$ in $Q(x)=\frac{6}{\pi^2}x+E(x)$, where $Q(x)$ counts the squarefree integers up to $x$.

• Number theory

Asks for the order of magnitude of Jacobsthal's function $h(k)$ over integers with at most $k$ distinct prime factors, and whether $h(k)\ll k^2$. An accepted partial claim proves the quadratic bound, and the order of magnitude stays open.

• Number theory

Asks whether, for irrational $\alpha>1$, infinitely many primes $p$ have $\lfloor p\alpha\rfloor$ prime; open. The one-prime statement is classical, and the two-prime statement is known only for almost all $\alpha$.

• Number theory

Asks whether, for an irreducible nonconstant $f\in\mathbb{Z}[x]$ with $f(n)\ge 1$ for large $n$, there is a constant $c=c(f)>0$ with $\sum_{n\le X}\tau(f(n))\sim cX\log X$.

• Number theory
• Divisors
• Polynomials

Asks for estimates of the greatest prime factor $F_f(n)$ of $\prod_{m\le n}f(m)$ for an irreducible polynomial $f$, in particular whether $F_f(n)\gg n^{1+c}$ for some $c>0$.

• Number theory

Asks whether, for each $k\geq 2$, the number of ways to write $n=p_1^k+\cdots+p_k^k$ with all $p_i$ prime is unbounded.

• Number theory

Asks whether $n$ points in convex position in the plane always include a vertex with at least $\lfloor n/2\rfloor$ distinct distances to the other vertices.

• Geometry
• Convexity
• Distances

Asks whether every prime $p>2$ has a primitive root modulo $p$ that is a prime $q<p$. The erdosproblems.com wording also includes $p=2$, where it fails.

• Number theory

Asks whether $\phi(n)=\phi(n+1)$ for infinitely many $n$, where $\phi$ is Euler's totient function.

• Number theory

Asks whether, for every fixed $c>0$ and all large $x$, some $n\le x$ has the values $\phi(n+k)$ distinct for $1\le k\le(\log x)^c$.

• Number theory

Asks for $f_r(n)$, the least number of edges that forces a triangle in a graph on $n$ vertices with chromatic number at least $r$. It is known exactly for $r\le3$ and for $r=4$ with $n$ large, and open in general.

• Graph theory

Asks for an asymptotic formula for $h_3(k)$, the fewest vertices of a triangle-free graph with chromatic number $k$, and for a proof that $h_3(k+1)/h_3(k)\to1$.

• Graph theory
• Chromatic number

Asks to estimate the number of edge-disjoint complete graphs needed to partition the edges of a graph on $n$ vertices with more than $n^2/4$ edges. It is open, though Győri and Keszegh settle the $K_4$-free case up to about $n^2/16$ edges beyond $n^2/4$.

• Graph theory

Asks whether, for $r\ge3$ and $n\ge rk$, the most edges in an $r$-uniform hypergraph on $n$ vertices with no $k$ pairwise disjoint edges is the larger of the clique count and the star count.

• Graph theory
• Hypergraphs

Asks whether $R(k)/(k2^{k/2})\to\infty$, where $R(k)$ is the Ramsey number of the complete graph $K_k$.

• Graph theory
• Ramsey theory

Asks whether the off-diagonal Ramsey number $R(k+1,k)$ exceeds the diagonal number $R(k,k)$ by a constant factor in the limit.

• Graph theory
• Ramsey theory

Asks whether, for arbitrarily large $n$, some $4$-chromatic critical graph on $n$ vertices has minimum degree $\gg n$. Such a graph has chromatic number $4$, and deleting any edge drops it to $3$.

• Graph theory
• Chromatic number

Asks whether there is a constant $c>0$ such that every graph on $2^n$ vertices with minimum degree greater than $(1-c)2^n$ contains the $n$-dimensional hypercube $Q_n$.

• Graph theory

Asks for the maximum of $\prod_{i\neq j}|z_i-z_j|$ over $z_1,\ldots,z_n\in\mathbb{C}$ with all $|z_i-z_j|\le2$, and whether a regular polygon attains it.

• Analysis

Asks whether $\sum_{n\ge1}1/(t^n-1)$ is irrational for every rational $t>1$.

• Irrationality

Asks whether there are only finitely many unitary perfect numbers, the integers $n$ equal to the sum of their proper divisors $d$ with $(d,n/d)=1$.

• Number theory

Asks whether every $k$ with $\sigma(n)=kn$ must satisfy $k=o(\log\log n)$.

• Number theory

Primes are sorted into classes by the prime factors of $p+1$, with class $1$ holding the primes $p$ for which $p+1$ has no prime factor other than $2$ and $3$. The problem asks whether every class is infinite and how $p_r^{1/r}$ behaves, where $p_r$ is the least prime in class $r$.

• Number theory
• Primes

Asks whether, for every $k\geq 2$, there are a prime $p$ and $k$ consecutive intervals of integers whose products are each congruent to 1 modulo $p$.

• Number theory

Asks whether the number $f(n)$ of solutions $k$ of $k\sigma(k)=n$ satisfies $f(n)\le n^{o(1/\log\log n)}$, or perhaps even $f(n)\le(\log n)^{O(1)}$.

• Number theory