33 problems

Let $r_2'(N)$ denote the number of representations of $N$ as a sum of two primes, and let $\mathfrak{S}(N)$ be the singular series … For even integers $N$, let $r_2(N)$ denote the…

Extended generalised Goldbach conjecture. If $m_2>0$, then every sufficiently large such $n$ has primes $p$ and $q$ satisfying

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…

Hua's conjecture. Every sufficiently large $n\in\mathcal H_2$ can be represented as the sum of a prime and the square of another prime, and every sufficiently large…

Romani's conjecture. The asymptotic density of this set is approximately $0.434$. This is a quantitative question about the exceptional structure of the Romanoff sumset; the source…

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…

Pure reversed-prime asymptotic conjecture. For every $A>0$,

Let $b\geq 2$ be fixed. Define … and … Also let $\#\mathfrak{B}(N)$ denote the number of admissible integer representations encoded by the source's set $\mathfrak{B}(N)$. Asymptoti…

Nnameir hypothesis. For sufficiently large $N$ and $B\geq 1$,

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 $C_1$ denote the constant governing the sieve estimates used to bound the lower density of integers representable as a prime plus a power of $2$. Elliott–Halberstam-related con…

For positive integers $n$, let … be the proportion representable as a prime plus a power of $2$, and let $d$ denote the corresponding limiting density, heuristically represented by…

For integers $n>1$, let $k,m$ be positive integers and suppose that $n=m+k$. Sun's $2^k+m$ conjecture. Every integer $n>1$ has such a representation for which $2^k+m$ is prime. Thi…

Let $k$ be a positive integer, and define … Here $(k,2)$ denotes the greatest common divisor of $k$ and $2$. The conjecture for $C_k$. The limit exists and satisfies … This follows…

Erdős's 1950 conjecture. There exists an integer $n$ such that the number of solutions of

Let $f_{2^{\mathbb P}}(n)$ denote the number of pairs $(p,a)$ such that $n=p+2^a$, where $p$ is prime and $a$ is prime. Chen's conjecture. The representation function is unbounded…

Three squares of primes conjecture. Every such $n$ can be represented as the sum of three squares of primes; that is, there exist primes $p_1,p_2,p_3$ such that

Bounded odd Gaussian-prime representation conjecture. There are constants $c_0$ and $k_0$ such that, for every $z\in\Gamma_G$ with…

Let $a,b$ be positive integers with $a>6$ and $a\geq b$. For a positive integer $k$, consider nonnegative integers $x_{1s},x_{2s}$ satisfying … and, for every $s$,…

Prime-power and two prime squares representation conjecture. Every sufficiently large $n$ subject to some congruence conditions can be represented as

Let $A=\left\{ n\in\mathbb{N}:\,n\equiv1\,\mod\,2;\,n\not\equiv2\,\mod\,3\right\}$. A natural number is represented as a sum of a prime and two prime squares if it has the form…

Finiteness conjecture. The sequence $s(n)$ contains only finitely many zero terms. This asserts that $p_n+k-3$ is represented by $F_k$ for some $k\geq3$ for all but finitely many p…

Let $R(n)$ denote the number of ordered pairs of odd primes $(p,q)$ satisfying $p+q=n$, and let … be the twin-prime constant. Hardy–Littlewood's conjecture. As $n$ tends to infinit…