89 problems

Goldbach-event independence conjecture. The events $A$ and $A'$ satisfy

Generalized Andrews identity.

Let $k\geq 9$, and let the $n$-th non-$k$-gonal pyramidal number be the $n$-th positive integer that is not a $k$-gonal pyramidal number. The preceding formula defines this sequenc…

The binary Goldbach conjecture asserts that every even integer greater than $2$ is the sum of two primes:…

$\forall n\in\mathbb{Z}$ with $n>2$ and $n\equiv 0\pmod{2}$, there exist primes $p,q$ such that $n=p+q$.

For every integer $n>14$, there exist an odd prime $p$ and positive integers $x,y$ such that $n=p+x^2+y^2$.

Let $\mathbb N=\{0,1,2,\ldots\}$. For an integer $k>1$, a set $A\subseteq\mathbb N$, and $n\in\mathbb N$, define $R_{1,k}(A,n)=\#\{(a_1,a_2)\in A^2:a_1+ka_2=n\}$. For integers…

For every integer $h\ge 3$ and every real number $\alpha$ with $0<\alpha<1/h$, does there exist a set $A\subseteq\mathbb{N}_0$ such that: (i)…

Generalization of Goldbach's conjecture. Let $Q$ be a subset of natural numbers whose distribution is similar to primes. There exists $N_Q\in\mathbb{N}$ such that, for any even int…

Let $\mathbb{P}$ denote the set of primes, and let $\mathbf{POL}$ be the set of even numbers that occur infinitely many times as differences of two consecutive primes. Polignac's c…

Let $\mathbb{P}$ denote the set of primes. An even number is an element of $2\mathbb{N}$. Kronecker's conjecture. Every even number can be expressed in infinitely many ways as the…

Let $b\geq 2$ be a fixed base. For a positive integer, write $\overleftarrow{n}$ for its digit reversal in base $b$. Then $p_1$ and $p_2$ are primes, and…

Let $\mathbb{P}$ denote the set of primes. An even number is an element of $2\mathbb{N}$. Maillet's conjecture. Every even number is the difference of two primes. This is a foundat…

Let $d\geq 3$ be an integer, let $N\in\mathbb N$, let $a_1,\ldots,a_N\in\mathbb C$, and write $\mathbb T^2$ for the two-dimensional torus. For $p\geq 2(d+1)$ and every…

Let $\mathscr{A}$ and $\mathscr{B}$ be two sets of positive integers. For a set $S$ of positive integers, write $S(x)=|S\cap[1,x]|$. Chen's Romanov-type conjecture. If there exists…

Krause–Zahlten conjecture. The equality

Density conjecture for sums of distinct powers. If every pair $a_1,a_2\in A$ satisfies $\gcd\{a_1,a_2\}=1$ and

Chen's conjecture. If $r_1,\ldots,r_t$ are positive integers with $r_1^{-1}+\cdots+r_t^{-1}\geq 1$, then $\mathcal{R}_{\mathcal{A}}\cap(\mathcal{R}_{\mathcal{A}}+2)$ has positive l…

Let $d\geq 2$ and $k\geq 1$. Let $A_i$ be positive integers, let $u_i,v_i$ be integers, let $\lambda\ne 0$ be a real constant, and let $L(M)$ be an integer-valued linear polynomial…

Let $d\geq 2$, $k\geq 1$, and $\varepsilon>0$. For $N\geq 1$ and $\beta\in\mathbb R/\mathbb Z$, consider the distance to the nearest integer, denoted by $\|\cdot\|$. Uniform expone…

For a fixed base $b\geq 2$, let $\overleftarrow{K}_b$ be the smallest number greater than $1$ such that every sufficiently large integer is a sum of at most $\overleftarrow{K}_b$ r…

For a fixed base $b\geq 2$, define the Nnamlerinchs constant $\overleftarrow{K}_b$ to be the smallest number greater than $1$ such that every sufficiently large integer can be writ…

Work in base $b=10$, and write $\overleftarrow{p}$ for the digit reversal of a prime $p$. Base-10 ternary reversed-prime conjecture. Every sufficiently large odd integer $N$ can be…

Let $b$ be a fixed prime, and let $\overleftarrow{p}$ denote the digit reversal of a prime $p$ in base $b$. Two-reversed-prime conjecture. Every sufficiently large even integer $N$…

Let $p_n$ denote a prime number, and for each integer $k\geq 1$ define … where $p_{n+i}$ are consecutive primes. Infinite admissible lengths conjecture. For every prime number…