44 problems

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

Let $\pi_2(x)$ count twin-prime pairs up to $x$, let $C_2$ be the twin-prime constant, and let $\operatorname{Li}_2(x)$ denote the corresponding second-order logarithmic integral.…

Let $m\geq4$, set $A_m=\{1,\ldots,m-3\}$, and define … with, for $n\geq2$, … Let $n^*=n^*(m)$ be the point of the last nontrivial increment of $\{\tilde{c}^{(m)}(n)\}$ on $A_m$, wi…

Let $\tilde{c}(n)$ be the sequence defined by … An increment is called main if it occurs at one of the distinguished fundamental points described in the paper. Shevelev's main-incr…

Let $p$ range over primes. Twin prime conjecture. There are infinitely many prime pairs $p$ and $p+2$. This is the case of the smallest possible gap between primes, apart from the…

Let $p_n$ denote the $n$th prime number. Twin-prime conjecture. … Equivalently, there are infinitely many primes $p$ such that $p+2$ is prime. The conjecture remains open; the sour…

Expected twin-prime gap conjecture. Wherever one happens to be in the infinite set of primes, it should be possible to characterize the expected number of primes encountered before…

Let $0<\theta<1$ be fixed, let $h\neq0$ be a fixed even integer, and let $\Lambda(n)$, $\mu(n)$ and $\varphi(q)$ denote the von Mangoldt function, Möbius function and Euler totient…

For a squarefree integer $Q$ with $6\mid Q$, let $U_Q$ be the set of residues $r\bmod Q$ such that both $r$ and $r+2$ are coprime to $Q$. Define the residue-pairing bound … where ……

Twin prime conjecture. This equation has infinitely many prime solutions as $x\to\infty$.

A prime is an integer greater than $1$ with no positive divisors other than $1$ and itself. The twin prime conjecture. There exist infinitely many primes $p$ such that $p+2$ is als…

Nicely's conjecture. Brun's constant satisfies

For each natural number $n$, let $p$ range over natural numbers and let $\operatorname{Prime}(p)$ mean that $p$ is prime. Twin prime conjecture. There are infinitely many pairs of…

Reciprocal-sum conjecture.

A pair of primes with gap $d$ is a pair of primes whose difference is $d$. The twin prime conjecture. There are infinitely many pairs of primes that differ by $2$. This remains a m…

Let $p$ range over the prime numbers. Twin prime conjecture. There are infinitely many primes $p$ such that $p+2$ is also prime. This is a classical unsolved problem in number theo…

Type-III bias conjecture. As $x\rightarrow\infty$,

Type-II bias conjecture. For any two residue classes $a_i$ and $a_j$,

Twin-prime progression conjecture. For every natural number $l$ with $l>2$, there exist arithmetic progressions of twin primes with $l$ terms.

A twin prime pair is a pair of primes differing by $2$. Twin prime conjecture. There are infinitely many twin primes. The conjecture is a classical open problem in number theory; t…

Universal primorial conjecture. There are expected to be infinitely many numbers $x=Kp_n\#+g$ such that both $x$ and $x+2$ are prime.

Lillie's conjecture. The total expected number of primorial twin prime pairs is approximately three.

Wolf's conjecture. The number of sign changes of $d_2(n)$ for $n\in(1,T)$ is $\frac{\sqrt T}{\log T}$. This numerical prediction concerns fluctuations around the Hardy–Littlewood e…

Quantitative twin-prime lower-bound conjecture. For each integer $a>0$ there exists an integer $N_2(a)$ such that

Twin prime conjecture. There exist infinitely many twin primes.