36 problems

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

Exponential bounding-functions conjecture. The lower and upper bounding functions of $G(n)$ can be expressed respectively in the form

A strong form of Goldbach's conjecture. For every even integer $n\geq 6$,

Let a finite Goldbach graph be a finite graph associated with Goldbach representations, and let a finite near Goldbach graph be a finite graph obtained from the corresponding near…

Let $n$ be an even Gaussian integer, let $N(n)$ denote its norm, and let $\arg(n)$ denote its argument. Gaussian Goldbach conjecture with angular restrictions. Every even Gaussian…

Generalised Goldbach conjecture. Under these conditions, there exist primes $p$ and $q$ such that

Goldbach-Twin conjecture. For any positive even natural number $n$, there are at least two prime numbers $p,q\in\mathcal{P}$ such that $m=p+q$ and $p$ or $q$ is a twin prime. The s…

Goldbach-Intervals conjecture. For every positive natural number $n$ and every $m\in\mathcal{E}_{n+1}$, there are at least two prime numbers $p\in\mathcal{P}_n$ and…

Let $\Lambda$ be the von Mangoldt function, let … and let $\mathfrak{S}(n)$ be the singular series defined by … for even $n$, with $C_2=\prod_{p>2}(1-(p-1)^{-2})$. The weakened Gol…

Let $k$ be a positive integer and let $C_{2,k}$ be the prime-pair constant. Extended Goldbach conjecture. The number of ways of writing $2k$ as a sum of two primes is asymptotic to…

Let $n$ be a positive even integer. A pair of Goldbach representations consists of primes $q_1,q_2,q_4,q_5$ satisfying … and … There is also a prime $q_3$ with $q_5<q_3<n$. Double…

Let a Goldbach factorization graph be a graph associated with an even integer and let an exceptional autonomous component (EAC) denote the special source strongly connected compone…

Let $A(p,q)$ be the arithmetic progression whose first two terms are odd primes $p$ and $q$, and let $n$ be a positive integer. Unified weak Goldbach conjecture. For each positive…

Weak odd Goldbach conjecture. For each odd positive integer $a>2$, there exist a positive integer $n$ and odd primes $p$ and $q$ such that

Weak even Goldbach conjecture. For each even positive integer $a>2$, there exist a positive integer $n$ and odd primes $p$ and $q$ such that

Let $p,q,r$ be primes. The ternary progression Goldbach conjecture. Every odd integer $n>5$ can be written as … with … This is stated as a ternary analogue of one of the explicit m…

Let $E_{a,b,m}$ be the exceptional set of positive even integers in the residue class $a+b\pmod m$ that do not have a representation $n=p+q$ with primes $p\equiv a\pmod m$ and…

Exceptional-set growth conjecture. As $m\to\infty$,

Let $p$ and $q$ be primes. The selected-moduli refined Goldbach conjecture. The following assertions hold: 1. Every even $n\equiv0\pmod3$ except $n=6$ has a representation with…

Let $p$ and $q$ be primes. The mod-$4$ refined Goldbach conjecture. The following assertions hold: 1. Every even $n>4$ has a representation $n=p+q$ with $p\equiv3\pmod4$. 2. Every…

Let $N$ be an integer, let $p$ range over the primes dividing $N$, and let $\Lambda_0$ denote the function used in the source. Prime-divisor positivity conjecture. If … then … for…

Let $p_{1},\ldots,p_{k}$ denote the first $k$ odd primes in succession, and let $n$ be an even integer such that $n>p_{k}$. Williamson's prime-factor conjecture. If, for all…

Let $n$ be an even positive integer. An odd prime is a prime number other than $2$. Asymptotic binary Goldbach and Lemoine conjecture. For every sufficiently large even number $n$,…

Tightened Goldbach conjecture. For all $n\geq 6$, there exist primes $q_1,q_2\in\mathbb{P}$ such that

Euler's totient formulation of the ternary Goldbach theorem with a congruence. The system has a solution $(x,y)$ in non-negative integers for every odd integer $n$ greater than fiv…