68 problems

Fermat prime finiteness conjecture. There are only finitely many Fermat primes, that is, only finitely many indices $n$ for which $F_n$ is prime. The heuristic is based on summing…

Bunyakovsky's conjecture. The value $f(n)$ should be prime for infinitely many positive integers $n$.

Barban–Davenport–Halberstam conjecture. Under these conditions,

Let $A\geq 2$, and let $b$ and $l$ be coprime integers. Heath-Brown's conjecture. There exists a prime $p\equiv b\pmod l$ such that … This conjecture is used as a conditional input…

Let $k$ be the length of the progression, let $d$ be its common difference, and let $A_{k,d}$ be the adjustment factor defined by the local prime-divisibility probabilities. Prime…

Let $S_0(a,b)$ be the numerical semigroup generated by $a$ and $b$, let $g_{0,a,b}$ be its Frobenius number, and let $\pi_{0,a,b}$ count the primes in $S_0(a,b)$ not exceeding…

Let $\mathcal{P}$ denote the set of primes, and let $1_{\mathcal{P}}$ be its indicator function. For a tuple of distinct integers ${\mathcal H}=\{h_1,\dots,h_k\}$, let…

Let $r_{p_0}$ be a ray with $p_0\in\overline{C}$. Let $l$ denote the number of correct decimal digits available in the floating-point arithmetic, and let $n^*(l)$ be a finite index…

Universal dominance conjecture. For any modulus $N\ge 3$, the quantity $\log L(1,\chi_{1,a})$ achieves its unique absolute minimum over all non-principal residue classes…

Let $0<\theta<1$ be fixed, let $N$ be a sufficiently large natural number, and let $g$ be either $g(n)=\mu(n)$ or $g(n)=\mu(n)\log(n)$. Let $\Lambda(n)$ and $\varphi(q)$ denote the…

Let $0<\theta<1$ be fixed, let $N$ be a sufficiently large natural number, and let $\Lambda(n)$, $\mu(n)$ and $\varphi(q)$ denote the von Mangoldt function, Möbius function and Eul…

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…

Let $A$ be a set whose elements are all products of an odd number of primes, or all products of an even number of primes. Parity problem. Without injecting additional ingredients,…

Chebyshev-like bias conjecture. (Weak) If $U$ is a regular Lucas sequence with $D>0$, then

Let $G_0(a,b)$ be the set of gaps of the numerical semigroup generated by $a$ and $b$, let $g_{0,a,b}$ be its Frobenius number, and let $\pi^*_{0,a,b}$ be the number of primes in…

Let $a$ and $b$ be integers with $b>a\ge 1$ and $\gcd(a,b)=1$. Set $S(a,b)=ab-a-b$, and let $\pi^*(a,b)$ be the number of primes not belonging to the numerical semigroup…

Model bounds conjecture. As $n\to\infty$, the estimates satisfy

Let $\vartheta$ denote the level of distribution of the shifted primes. Level of distribution one conjecture. The shifted primes have level of distribution … A level of distributio…

Let $0<\theta\leq 1$ be real, and let $\pi_{\theta}(x)$ denote the number of integers $n$ with $1\leq n\leq x^{\theta}$ such that $\left\lfloor x/n\right\rfloor$ is prime. For a gi…

Let $f_1(t)=a_1t+b_1$ and $f_2(t)=a_2t+b_2$ be linear forms with integers $a_1,a_2,b_1,b_2$ and $a_1,a_2>0$. Set $f=f_1f_2$, and suppose that $f$ is admissible, meaning that it doe…

Let $\mathcal P(g,t)$ and $\mathcal Q_{\alpha,\beta}$ be the prime sets defined earlier in the paper, and let $\delta(g,t)$ and $\rho_{\alpha,\beta}(a,d)$ be the corresponding cons…

Let $R_B(x)$ denote the number of $B$-regular primes up to $x$, where an odd prime is $B$-regular if it divides none of the numerators of $B_2,B_4,\ldots,B_{p-3}$. Siegel's conject…

Let $f(x,y)$ be a homogeneous polynomial of degree $n$, let $F(t)=\int_0^t[\sqrt[n]{f(1,y)}]^{-2}\,dy$, and let $\mathfrak{a}$ be an ideal in $\mathbb{Q}[\xi]$, where $\xi$ is a ro…

For imaginary quadratic fields, let $P_M(-d)$ count primes less than $M$ that split completely and whose factors lie in the cone $C_1$. Let…

Let $K=\mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic field, and let $C_1$ be the cone whose elements have angle between $\pi/36$ and $2\pi/36$. Let $P_M(-d)$ count primes less t…