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

Ordowski's density conjecture. The set $\mathcal S$ has an asymptotic density. Counts up to $10^8$ suggest that this density may be about $5/8$. The source proves that the asymptot…

Residue-class conjecture. These conditions are sufficient for there to be infinitely many base-$a$ pseudoprimes $n\equiv r\pmod m$; explicitly, they require

A Baillie–PSW pseudoprime is an odd composite integer that passes both the strong probable-prime test and the Lucas primality test. Baillie–PSW pseudoprime infinitude conjecture. T…

Let $f(x)\in\mathbb{Z}[x]$ be a monic, squarefree polynomial that is not the product of cyclotomic polynomials. Let $N_f(y)$ denote the number of Frobenius pseudoprimes with respec…

Let $M\geq 7$. Choose distinct primes $p,q$ uniformly at random subject to $5\leq p,q\leq M$, and set $N=pq$. Choose an elliptic curve $E(\mathbb{Z}/N\mathbb{Z})$ uniformly at rand…

Let the three propositions above be the results concerning the sets and quantities introduced in this section, each currently stated for a restricted range of $calpha$. Exponent co…

Novák's sparsity conjecture. For every $\varepsilon>0$,