26 problems

Let $e,f,e',f'$ be integers with $1\le e\le f$, $e+f\ge 3$, and likewise $1\le e'\le f'$ and $e'+f'\ge 3$. Let ${\mathcal P}_w(e,f)$ denote the set of pairs $(p,q)$ of distinct pri…

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 $n$ 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…

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.

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

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…

Density-minimum conjecture.

Let $C(x)$ denote the number of Carmichael numbers $n\leq x$ with exactly three prime factors. Three-factor Carmichael number conjecture. As $x\to\infty$, one has … where $C$ is an…

A super Carmichael number is a weak Carmichael number satisfying … Super-Carmichael containment conjecture. Every super Carmichael number is a Carmichael number. The source present…

A super Carmichael number is a weak Carmichael number satisfying … Super Carmichael infinitude conjecture. There are infinitely many super Carmichael numbers. The source motivates…

A super Carmichael number is a weak Carmichael number satisfying … No odd prime-power super Carmichael conjecture. For every odd prime $p$ and every $f\ge 2$, the prime power $p^f$…

For a composite integer, call the number of distinct prime divisors its number of prime factors. Smallest weak Carmichael number conjecture. For every integer $k\ge 3$, the smalles…

Two Carmichael numbers are twin Carmichael numbers if no weak Carmichael number lies strictly between them. Twin Carmichael numbers conjecture. There are infinitely many pairs of t…

Let $C(n)$ and $W(n)$ denote the numbers of Carmichael numbers and weak Carmichael numbers in the interval $[1,n]$, respectively. Asymptotic equality conjecture. … The conjecture p…

Let $s\ge 2$ be an integer, and let $(e_1,e_2,\ldots,e_s)$ be a fixed tuple with $e_1\ge e_2\ge\cdots\ge e_s\ge 1$ and $\sum_{i=1}^s e_i\ge 3$. Generalized weak Carmichael infinitu…

For integers $e,f$ with $1\le e\le f$ and $e+f\ge 3$, and an odd prime $q$, let ${\mathcal Q}_w(q;e,f)$ be the set of primes $p$ such that $p^eq^f$ or $q^ep^f$ is a weak Carmichael…

For integers $e,f$ with $1\le e\le f$ and $e+f\ge 3$, let ${\mathcal Q}_w(e,f)$ be the set of primes occurring in a weak Carmichael number of the form $p^eq^f$ or $q^ep^f$ for some…

Let $M$ be a positive integer, and let $a$ be an integer such that $(a,M)=1$. A Carmichael numbers in arithmetic progressions conjecture. There are infinitely many Carmichael numbe…

Let $K(x)$ be the number of integers up to $x$ satisfying the paper's radically weakened Carmichael condition, and let $C(x)$ be the number of Carmichael numbers up to $x$. Growth-…