Asks whether, for every $C\ge 0$, some sequence of indices $n_i$ has $(p_{n_i+1}-p_{n_i})/\log n_i\to C$, where $p_n$ is the $n$th prime.

• Number theory
• Primes

Asks whether there is a covering system of congruences whose moduli are distinct and all odd.

• Number theory
• Covering systems

Asks whether the odd integers not of the form $p+2^k+2^l$, with $p$ prime and $k,l\geq 0$, have positive upper density.

• Number theory
• Additive bases
• Primes

Asks whether some fixed $k$ makes every large integer the sum of a prime and at most $k$ powers of 2. It is open, but three powers are known not to suffice: infinitely many even integers are not a prime plus at most three powers of 2.

• Number theory
• Additive bases
• Primes

Asks whether every large odd integer is the sum of a squarefree number and a power of 2.

• Number theory
• Additive bases

Asks how large an infinite set with no element dividing the sum of two larger elements can be, in terms of its counting function, its density and its reciprocal sum. The first two questions are answered by a refereed construction of 2026, and the reciprocal-sum question is open.

• Number theory

Asks whether $\sum_{n\ge 1}(-1)^n n/p_n$ converges, where $p_n$ is the $n$th prime; open. Tao proved convergence conditional on a strong Hardy-Littlewood prime tuples conjecture.

• Number theory
• Primes

Asks whether infinitely many primes $p$ have every even $n\le p-3$ expressible as $q_1-q_2$ with primes $q_1,q_2\le p$.

• Number theory
• Primes

Asks how small $h(m)$ can be, the number of distinct divisors of a practical number $m$ that always suffice to write each smaller integer as a sum, especially for $m=n!$. The bound $h(n!)<n^{o(1)}$ is proved (Conjectures.io, 2026), the $(\log\log m)^{O(1)}$ part is claimed (van Doorn and GPT-6 Astra Pro, 2026, graded sound by Plasma AI's internal review, its sign-off pending), and the rest is open.

• Number theory
• Divisors
• Factorials

Asks whether every edge-disjoint union of $n$ copies of $K_n$ has chromatic number exactly $n$.

• Graph theory
• Chromatic number

Asks whether every triangle-free graph on $5n$ vertices can be made bipartite by deleting at most $n^2$ edges.

• Graph theory

Asks whether the integers avoiding a chosen residue class for each modulus in an increasing sequence always have a logarithmic density.

• Number theory

Asks whether, when $A+A$ contains all but finitely many integers, some integers must have arbitrarily many representations as a sum of two elements of $A$.

• Number theory
• Additive bases

Asks whether the largest Sidon set in $\{1,\ldots,N\}$ has size $N^{1/2}+O_\epsilon(N^\epsilon)$ for every $\epsilon>0$.

• Number theory
• Sidon sets
• Additive combinatorics

Asks how sparse a set $A$ can be if every large integer is a prime plus an element of $A$, measured against $(\log N)^2$.

• Number theory
• Additive bases

Asks how sparse a set $A$ can be if every large integer is a square plus an element of $A$, measured against $N^{1/2}$.

• Number theory
• Additive bases

Asks for the largest constant $c$ such that, for large $N$, every partition of $\{1,\ldots,2N\}$ into sets $A$ and $B$ of size $N$ has some $x$ with at least $cN$ solutions to $a-b=x$ with $a\in A$ and $b\in B$.

• Number theory
• Additive combinatorics

Asks whether some infinite Sidon set has $\gg_\epsilon N^{1/2-\epsilon}$ elements up to $N$ for every $\epsilon>0$.

• Number theory
• Sidon sets
• Additive combinatorics

Asks for which functions $g(N)\to\infty$ a set with $\gg N^{1/2}/g(N)$ elements up to $N$ must have integers with arbitrarily many representations as a sum of two elements.

• Number theory
• Additive bases

Asks whether an infinite set $A$ whose triple sums $a+b+c$ are all distinct must have $\liminf \lvert A\cap\{1,\ldots,N\}\rvert/N^{1/3}=0$.

• Number theory
• Sidon sets
• Additive combinatorics

Asks whether, for any $\epsilon>0$, every Sidon set $A\subset\{1,\ldots,N\}$ extends by elements above $N$ to a Sidon set of size at least $(1-\epsilon)M^{1/2}$ in some $\{1,\ldots,M\}$.

• Number theory
• Sidon sets
• Additive combinatorics

Asks whether the density $f(c)$ of the integers $n$ with $\phi(n)<cn$, as a function of $c$, ever has a positive derivative.

• Number theory

Asks whether some infinite set $A$ of totient values has $n_a/a\to\infty$, where $n_a$ is the least $n$ with $\phi(n)=a$.

• Number theory

Asks whether every finite set $A$ of integers has $\max(\lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon\lvert A\rvert^{2-\epsilon}$ for every $\epsilon>0$.

• Number theory
• Additive combinatorics

Asks whether every graph on $n$ vertices with more than $\mathrm{ex}(n;C_4)$ edges contains $\gg n^{1/2}$ copies of $C_4$.

• Graph theory
• Cycles