53 problems

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

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

Let $r(n)$ denote the number of ways an even integer $n=2m$ can be expressed as a sum of two primes. Let $\pi(n)$ denote the number of primes up to $n$, and let the product below e…

Prime-gap bound conjecture. With $\beta=10^{7194}$, the displayed inequality holds, and consequently the ternary Goldbach conjecture follows by the argument given.

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 $n\geq 2$ be an integer. A symmetric representation of $2n$ as a sum of two primes has the form … for primes $p_1,p_2$ and an integer $j\geq 0$. Goldbach's symmetric-center for…

Hardy–Littlewood's generalised Goldbach conjecture. The assertion is that every sufficiently large $n$ meeting these congruence conditions has an $(m_1,m_2)$-partition into two pri…

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…

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

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…

Let $\mathcal O_{\mathbb Q(\sqrt{-1})}$ and $\mathcal O_{\mathbb Q(\sqrt{-3})}$ denote the rings of integers of the Gaussian and Eisenstein quadratic fields, respectively. Define ……

Granville's growth conjecture.

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 $h$ be an integer with $h\geq 4$, let $\Lambda$ denote the von Mangoldt function, and let $\mathfrak{S}_h$ denote the Goldbach singular series. Asymptotic Goldbach conjecture.…

Let $\mathbb{N}_{\mathsf{G}}$ be the set of Googolplex-bounded numbers, namely natural numbers less than $10^{10^{100}}$. Googolplex Goldbach conjecture. Every even Googolplex-boun…

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…

Vertical Generalization of the binary Goldbach conjecture. For each entry $L_{(a,b)}$ of $L$, every even integer satisfying

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 $n$ be an integer and let $2n\geq 8$ be even. The primes $p$ and $q$ are distinct odd primes when both are odd prime numbers and $p\ne q$. Strongly binary Goldbach conjecture.…

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