45 problems

Carmichael-number counting conjecture. The count of Carmichael numbers up to $x$ is of the form $x/L(x)^{1+o(1)}$ as $x\to\infty$. The known upper bound is $C(x)\le x/L(x)$ for all…

Let $C(X,1)$ denote the number of Carmichael numbers up to $X$. Pomerance's conjecture. For sufficiently large $X$, … This conjectural lower bound is compared with Pomerance's prov…

Michon's conjecture. Every odd cyclic number has at least one Carmichael multiple.

Infinitude conjecture for rigid Carmichael numbers. There are infinitely many rigid Carmichael numbers of order $d$ for any $d\geq 2$.

Let $n$ be a composite almost-prime number, and write $\omega(n)$ for the number of its prime divisors. Distribution conjecture. There exists an $\epsilon<1$ such that … is satisfi…

A positive integer $n$ is almost-prime if it is square-free and satisfies $n\mid T_n(x)$ for every integer $x$, where … and $1=d_1<d_2<\dots<d_k=n$ are all divisors of $n$. Infinit…

For each positive integer $n$, let … where $1=d_1<d_2<\dots<d_k=n$ are all divisors of $n$. A positive integer $n$ is weakly almost-prime if $n\mid T_n(x)$ for every integer $x$; i…

Let $n$ be an odd squarefree composite integer, and let Euler liars in $\mathbb{Z}_n^*$ mean the elements $a$ satisfying the Euler probable-prime congruence…

Let $C(x)$ denote the number of Carmichael numbers at most $x$, and let the upper bound for $C(x)$ be the one stated in Theorem. Sharp Carmichael-count conjecture. The upper bound…

Let $n$ be an integer tested by the algorithm described in the paper, using Fermat and strong Fermat tests and the resulting splittings. Carmichael-recognition algorithm conjecture…

Let $n$ be a positive integer. The query whether $n$ is a Carmichael number is the decision problem of determining whether $n$ is composite and satisfies the defining Carmichael pr…

Hybrid method pair-count conjecture. The hybrid method considers

Carmichael's fixed-prime-factor infinitude conjecture. For any $R\in\mathbb N$ with $R\geq 3$, there exist infinitely many Carmichael numbers with $R$ prime factors.

Prime tuples conjecture. Every finite set of linear forms can be simultaneously prime for infinitely many integer values of $n$, unless the set is inadmissible.

Let $N_1(X)$ denote the number of Carmichael numbers less than or equal to $X$. Erdős's refined conjecture. From a series of assumptions, Erdős conjectured that … This is a more pr…

Let $N_1(X)$ denote the number of Carmichael numbers less than or equal to $X$. Equivalently, $N_1(X)$ is the counting function for the set $N_1$ of Carmichael numbers. Erdős's con…

Generalized Carmichael symmetry conjecture. For all integers $k$,

A Carmichael number is a positive composite integer $n$ such that $a^{n-1}\equiv 1 \pmod{n}$ for every integer $a$ with $1<a<n$ and $\gcd(a,n)=1$. Carmichael's conjecture. There ar…

For $x>0$, let $C_3(x)$ denote the number of three-factor Carmichael numbers at most $x$, and let $C'_3(x)$ denote the number of primary three-factor Carmichael numbers at most $x$…

Infinitude conjecture for the sets of digit-sum classes. For each $d\geq2$, the sets $\mathcal{S}_d$ and $\mathcal{S}\setminus\mathcal{S}_d$ are infinite. Moreover,…

Growth conjecture for primary Carmichael numbers. For sufficiently small $\varepsilon>0$,

Infinitude conjecture for primary Carmichael numbers. The following claims are true: the set $4\mathcal{C}'4$ is infinite, and the set $4\mathcal{C}\setminus\mathcal{C}'4$ is infin…

McNew's conjecture. There are infinitely many radimichael numbers which are not Carmichael numbers.

Let $a,b\in\mathbb{Z}$ satisfy $(a,b)=1$. An arithmetic progression with these parameters is the set of integers $a+bc$ with $c\in\mathbb{Z}$. Banks–Pomerance conjecture. There are…

Density-minimum conjecture.