175 problems

Let $p_n$ denote the $n$th prime. Cramér's conjecture. … This is a central conjecture on maximal prime gaps. The source notes that Granville challenged the constant $1$ in light of…

Let $n$ be an integer with $n\ge 1$. A prime number is an integer greater than $1$ with no positive divisors other than $1$ and itself. Legendre's conjecture. For each integer…

Let $p_n$ denote the $n$th prime. Firoozbakht's conjecture. For every $n$, … The conjecture implies increasingly strong upper bounds on prime gaps, including the bounds quoted in t…

Let $p_n$ denote the $n$th prime. Andrica's conjecture. For every $n$, … This conjecture gives a strong upper bound on gaps between consecutive primes and remains open.

Let $n$ be a positive integer. Oppermann's conjecture. There is a prime between $n(n-1)$ and $n^2$, and a prime between $n^2$ and $n(n+1)$. This stronger assertion remains open and…

Let $p_n$ denote the $n$th prime, with $n\ge 2$. Brocard's conjecture. There are at least four primes between $p_n^2$ and $p_{n+1}^2$. This is an open conjecture on the number of p…

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 $\mathbb{P}$ 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…

For $t>0$, let the jumping champion be the integer $g$ occurring most frequently as a gap between two successive primes less than or equal to $t$. Jumping champions conjecture. The…

Odlyzko–Rubinstein–Wolf conjecture. The jumping champions tend to infinity. Furthermore, any fixed prime $p$ divides all sufficiently large jumping champions.

Let $\mathcal{P}_{\mathrm{good}}=\{g_1,g_2,\ldots\}$ be the subset of good primes, enumerated in increasing order, where good primes are the primes outside the paper's exceptional…

Kanold's conjecture. One has

Wolf's conjecture. As $x\to\infty$,

Let $x$ and $y$ be positive real numbers, and consider intervals of length $y$ whose location is near $x$. Let the maximum be taken over such intervals of the number of primes they…

Let $p_n$ be the $n$-th prime, with $p_0=1$, and let … be the first $n$ prime gap sequence. A prime gap graph on $n\geq 2$ vertices is a simple graph whose vertex degrees are exact…

Let $p_j$ denote the $j$th prime and let $\pi(x)$ be the number of primes not exceeding $x$. Cramér-Shanks conjecture. As $x\to\infty$, … This is the classical prediction that the…

Let $\mathcal{D}_k^1=L_{\infty}(p_{k+1})-L_{\infty}(p_k)$ be the prime gaps along the number trail, and let $\mathbb{N}$ denote the natural numbers. Modified Polignac's conjecture.…

Let $p_n$ denote the $n$th prime. Prime-gap conjecture below the square-root scale. For every $n\ge31$ (so that $p_n\ge127$), … The paper states that this conjecture implies its cy…

Maier–Pomerance conjecture. Maier and Pomerance conjectured that

Let $G(x)$ denote the maximal gap between consecutive primes up to $x$. Shanks's conjecture. The maximal prime gaps satisfy … The source describes this as stronger than Cramér's li…

Polylogarithmic prime-gap conjecture. There exists an absolute constant $k>1$ such that

For each positive integer $n$, consider the consecutive squares $n^2$ and $(n+1)^2$. Conjecture on primes between consecutive squares. There is always a prime strictly between…

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…

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