相关论文: The Carmichael numbers up to $10^{18}$

We extend our previous computations to show that there are 246683 Carmichael numbers up to $10^{16}$. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a ``large prime…

We extend our previous computations to show that there are 585355 Carmichael numbers up to $10^{17}$. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a ``large prime…

We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with k prime factors for every k between 3 and 19,565,220. These computations are the product of implementations of two…

Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding three-prime…

For every sufficiently large integer $R$, there exists a Carmichael number with exactly $R$ prime factors.

Let $m$ be a Carmichael number and let $L$ be the least common multiple of $p-1$, where $p$ runs over the prime factors of $m$. We determine all the Carmichael numbers $m$ with a Fermat prime factor such that $L=2^{\alpha}P^2$, where $k\in…

Under the assumption of Heath-Brown's conjecture on the first prime in an arithmetic progression, we prove that there are infinitely many Carmichael numbers $n$ such that the number of prime factors of $n$ is prime.

We prove that there exist infinitely many (-1,1)-Carmichael numbers, that is, square-free, composite integers n such that p+1 divides n-1 for each prime p dividing n.

Our primary concern is the computational complexity of algorithms that find all Carmichael numbers less than some specified bound $B$. We have three related results. First, we show CARMICHAELS is in $\textbf{P}$, where only the run-time is…

Using the sieve, we show that there are infinitely many Carmichael numbers whose prime factors all have the form $p = 1 + a^2 + b^2$ with $a,b \in{\mathbb Z}$.

Erd\H{o}s conjectured in 1956 that there are $x^{1-o(1)}$ Carmichael numbers up to $x$. Pomerance made this conjecture more precise and proposed that there are $x^{1-{\frac{\{1+o(1)\}\log\log\log x}{\log\log x}}}$ Carmichael numbers up to…

We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZ-algebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine…

In this paper, we study the properties of Carmichael numbers, false positives to several primality tests. We provide a classification for Carmichael numbers with a proportion of Fermat witnesses of less than 50%, based on if the smallest…

For a Carmichael number $n$ with prime factors $p_1,\cdots,p_m$, define $$K=GCD[p_1-1,\cdots,p_m-1],$$ and let $C_\nu(X)$ denote the number of Carmichael numbers up to $X$ such that $K=\nu$. Assuming a strong conjecture on the first prime…

Fix coprime natural numbers $a,q$. Assuming the Prime $k$-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each of which lies in the reduced residue class $a$ mod $q$ and is a…

We report that there are $49679870$ Carmichael numbers less than $10^{22}$ which is an order of magnitude improvement on Richard Pinch's prior work. We find Carmichael numbers of the form $n = Pqr$ using an algorithm bifurcated by the size…

We define a variant of the Miller-Rabin primality test, which is in between Miller-Rabin and Fermat in terms of strength. We show that this test has infinitely many "Carmichael" numbers. We show that the test can also be thought of as a…

In this paper, we prove that for any $a,M\in \mathbb N$ with $(a,M)=1$, there are infinitely many Carmichael numbers $m$ such that $m\equiv a$ mod $M$

Alford, Granville, and Pomerance proved that there are infinitely many Carmichael numbers. In the same paper, they ask if a statement analogous to Bertrand's postulate could be proven for Carmichael numbers. In this paper, we answer this…

Improving on some recent results of Matom\"aki and of Wright, we show that the number of Carmichael numbers to $X$ in a coprime residue class exceeds $X^{1/(6\log\log\log X)}$ for all sufficiently large $X$ depending on the modulus of the…