We give an upper bound for the distribution of base-a pseudoprimes that is uniform in the base and does not require coprimality to the base. In addition we show that there are infinitely many “near Carmichael numbers” meaning that they are pseudoprimes for a positive proportion of bases, but not all bases.

Abstract The Piatetski-Shapiro sequences are sequences of the form ( ⌊ n c ⌋ ) n = 1 ∞ {\left(\lfloor {n}^{c}\rfloor )}_{n=1}^{\infty } and the Beatty sequence is the sequence of integers ( ⌊ α n + β ⌋ ) n = 1 ∞ {(\lfloor \alpha n+\beta \rfloor )}_{n=1}^{\infty } . We prove that there are infinitely many Carmichael numbers composed of entirely the primes from the intersection of a Piatetski-Shapiro sequence and a Beatty sequence for c ∈ 1 , 19137 18746 c\in \left(\phantom{\rule[-0.75em]{}{0ex}},1,\frac{19137}{18746}\right) , α > 1 \alpha \gt 1 irrational and of finite type by investigating the Piatetski-Shapiro primes in arithmetic progressions in a Beatty sequence. Moreover, we also discuss the intersection of a Piatetski-Shapiro sequence and multiple Beatty sequences in arithmetic progressions.

We report that there are 49679870 Carmichael numbers less than 1022 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 of P with respect to the tabulation bound B. For P<7×107, we found 35985331 Carmichael numbers and 1202914 of them were less than 1022. When P>7×107 7 \ imes 10^7$$\\end{document}]]>, we found 48476956 Carmichael numbers less than 1022. We provide a comprehensive overview of both cases of the algorithm. For the large case, we show and implement asymptotically faster ways to tabulate compared to the prior tabulation. We also provide an asymptotic estimate of the cost of this algorithm. It is interesting that Carmichael numbers are worst case inputs to this algorithm. So, providing a more robust asymptotic analysis of the cost of the algorithm would likely require resolution of long-standing open questions regarding the asymptotic density of Carmichael numbers.

Abstract This study explores fundamental aspects of probability theory within the framework of constructing random sequences or random number generators. We propose an interpretation of probability spaces and operations on formal events through the theory of formal languages, utilizing string manipulation techniques. As part of the research, we present a direct implementation of the discussed concepts in the form of a program that generates random numbers of the required type by processing signals from a sound card. Additionally, the problem of primality testing, which is particularly relevant to practical cryptographic applications, is addressed. We critically examine common misconceptions regarding the properties of Carmichael numbers and the application of Fermat’s Little Theorem. Furthermore, we propose an efficient primality testing algorithm.

Progress towards a conjecture of S.W. Graham

The current stage of scientific and technological development entails ensuring information security across all domains of human activity. Confidential data and wireless channels of remote control systems are particularly sensitive to various types of attacks. In these cases, various encryption systems are most commonly used for information protection, among which large prime numbers are widely utilized. The subject of research involves methods for generating prime numbers, which entail selecting candidates for primality and determining the primality of numbers. The subject of research involves methods for generating prime numbers, which choice selecting candidates for primality and determining the primality of numbers. The objective of the work is the development and theoretical justification of a method for determining the primality of numbers and providing the results of its testing. The aim to address the following main tasks: analyze the most commonly used and latest algorithms, methods, approaches, and tools for primality testing among large numbers; propose and theoretically justify a method for determining primality for large numbers; and conduct its testing. To achieve this aim, general scientific methods have been applied, including analysis of the subject area and mathematical apparatus, utilization of set theory, number theory, fields theory, as well as experimental design for organizing and conducting experimental research. The following results have been obtained: modern methods for selecting candidates for primality testing of large numbers have been analyzed, options for generating large prime numbers have been considered, and the main shortcomings of these methods for practical application of constructed prime numbers have been identified. Methods for determining candidates for primality testing of large numbers and a three-stage method for testing numbers for primality have been proposed and theoretically justified. The testing conducted on the proposed primality determination method has demonstrated the correctness of the theoretical conclusions regarding the feasibility of applying the proposed method to solve the stated problem. Conclusions. The use of a candidate primality testing strategy allows for a significant reduction in the number of tested numbers. For numbers of size 200 digits, the tested numbers is reduced to 8.82%. As the size of the tested numbers increases, their quantity will decrease. The proposed method for primality testing is sufficiently simple and effective. The first two stages allow for filtering out all composite numbers except for Carmichael numbers. In the first stage, using the first ten prime numbers filters out over 80 percent of the tested numbers. In the second stage, composite numbers with factors greater than 29 are sieved out. In the third stage, Carmichael numbers are sieved out. The test is polynomial, deterministic, and unconditional.

A Revisit of Carmichael Numbers and a Note on Carmichael Triplets

Previous work established the set of square-free integers n with at least one factorization n = p ¯ q ¯ for which p ¯ and q ¯ produce valid RSA keys, whether they are prime or composite. These integers are exactly those with the property λ ( n = p ¯ q ¯ ) ∣ ( p ¯ - 1 ) ( q ¯ - 1 ) , where λ is the Carmichael totient function. We refer to these integers as idempotent , because ∀ a ∈ Z n , a k ( p ¯ - 1 ) ( q ¯ - 1 ) + 1 ≡ n a for any positive integer k . This set includes the semiprimes and the Carmichael numbers, but is not limited to them. Numbers in this last category have not been previously analyzed in the literature. We discuss the structure of idempotent integers here, and present heuristics to assist in finding them. We introduce the notions of partial idempotency and minimal idempotency , give appropriate definitions for them, and present preliminary results.

In this paper we improve on previous bounds for the reciprocal sum of pseudoprimes and for the reciprocal sum of Carmichael numbers. In particular, we prove that these sums are less than 0.0911 0.0911 and 0.0058 0.0058 , respectively.

The Korselt numbers and sets were discussed for the first time in 2007. The problem can be considered as a new one with limited literature making it as a new field of research. Let N be a positive integer and α a non-zero integer. If N ≠ α and p divides N for each prime divisor p of N, then N is called an α−Korselt number (Kα-number). In this thesis, many concepts such as Korselt numbers that are related to Carmichael numbers have been studied. Korselt determined Korselt numbers by studying the converse of Fermat’s Little theorem and it can be noticed that all these numbers depend on number theory, prime numbers, divisibility and modular arithmetic. To validate the concerned theorems, an illustrated proofs were followed through detailed steps in addition to many examples are solved in order to support the correctness of these theories. It is important to say that some errors in literature were addressed by us. Consequently, we introduced proper corrections for them. Finally, many notes have been taken and directed us to build and develop a number of complicated algorithms, some of them in order to find Korselt sets for relatively large numbers in an effective way in a short time which may require a great time and need tedious effort if it is to be calculated manually

Motivated by Carmichael numbers, we say that a finite ring $R$ is a Carmichael ring if $a^{|R|}=a$ for any $a \in R$. We then call an ideal $I$ of a ring $R$ a Carmichael ideal if $R/I$ is a Carmichael ring, and a Carmichael element of $R$ means it genera

Let $R$ be a ring with identity, $\mathcal{U}(R)$ the group of units of $R$ and $k$ a positive integer. We say that $a\in \mathcal{U}(R)$ is $k$-unit if $a^k=1$. Particularly, if the ring $R$ is $\mathbb{Z}_n$, for a positive integer $n$, we will say that $a$ is a $k$-unit modulo $n$. We denote with $\mathcal{U}_k(n)$ the set of $k$-units modulo $n$. By $\text{du}_k(n)$ we represent the number of $k$-units modulo $n$ and with $\text{rdu}_k(n)=\frac{\phi(n)}{\text{du}_k(n)}$ the ratio of $k$-units modulo $n$, where $\phi$ is the Euler phi function. Recently, S. K. Chebolu proved that the solutions of the equation $\text{rdu}_2(n)=1$ are the divisors of $24$. The main result of this work, is that for a given $k$, we find the positive integers $n$ such that $\text{rdu}_k(n)=1$. Finally, we give some connections of this equation with Carmichael's numbers and two of its generalizations: Kn\"odel numbers and generalized Carmichael numbers.

We revisit the problem of tabulating Carmichael numbers. Carmichael numbers have been tabulated up to \(10^{21}\) using an algorithm of Pinch (Math Comp 61(203):381–391, 1993). In finding all Carmichael numbers with d prime factors, the strategy is to first construct pre-products P with \(d-2\) prime factors, then find primes q and r such that Pqr satisfies the Korselt condition. We follow the same general strategy, but propose an improvement that replaces an inner loop over all integers in a range with a loop over all divisors of an intermediate quantity. This gives an asymptotic improvement in the case where P is small and expands the number of cases that may be accounted as small. In head-to-head timings this new strategy is faster over all pre-products in a range, but is slower on prime pre-products. A hybrid approach is shown to improve even the case of prime pre-products.

Abstract Alford et al. [1] 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 question, proving the stronger statement that for all $\delta&amp;gt;0$ and $x$ sufficiently large in terms of $\delta $, there exist at least $e^{\frac {\log x}{(\log \log x)^{2+\delta }}}$ Carmichael numbers between $x$ and $x+\frac {x}{(\log x)^{\frac {1}{2+\delta }}}$.

We consider a generalization of Piatetski–Shapiro sequences in the sense of Beatty sequences, which is of the form $(\lfloor \alpha n^c + \beta \rfloor)_{n=1}^{\infty}$ with real numbers $\alpha \geq 1$, $c \gt 1$ and $\beta$. We show there are infinitely many primes in the generalized Piatetski–Shapiro sequence with $c \in (1,14/13)$. Moreover, we prove there are infinitely many Carmichael numbers composed entirely of the primes from the generalized Piatetski–Shapiro sequences with $c \in (1,64/63)$.

Computing the reciprocal sum of sparse integer sequences with tight upper and lower bounds is far from trivial. In the case of Carmichael numbers or twin primes even the first decimal digit is unknown. For accurate bounds the exact structure of the sequences needs to be unfolded. In this paper we present explicit bounds for the sum of reciprocals of Proth primes with nine decimal digit precision. We show closed formulae for calculating the nth Proth number F_n, the number of Proth numbers up to n, and the sum of the first n Proth numbers. We give an efficiently computable analytic expression with linear order of convergence for the sum of the reciprocals of the Proth numbers involving the Psi function (the logarithmic derivative of the gamma function). We disprove two conjectures of Zhi-Wei Sun regarding the distribution of Proth primes.

Previous work established the set of square-free integers n with at least one factorization n=p¯q¯ for which p¯ and q¯ are valid RSA keys, whether they are prime or composite. These integers are exactly those with the property λ(n)∣(p¯−1)(q¯−1), where λ is the Carmichael totient function. We refer to these integers as idempotent, because ∀a∈Zn,ak(p¯−1)(q¯−1)+1≡na for any positive integer k. This set was initially known to contain only the semiprimes, and later expanded to include some of the Carmichael numbers. Recent work by the author gave the explicit formulation for the set, showing that the set includes numbers that are neither semiprimes nor Carmichael numbers. Numbers in this last category had not been previously analyzed in the literature. While only the semiprimes have useful cryptographic properties, idempotent integers are deserving of study in their own right as they lie at the border of hard problems in number theory and computer science. Some idempotent integers, the maximally idempotent integers, have the property that all their factorizations are idempotent. We discuss their structure here, heuristics to assist in finding them, and algorithms from graph theory that can be used to construct examples of arbitrary size.

Arithmetic progressions of Carmichael numbers in a reduced residue class

We generalize the notions of elliptic pseudoprimes and elliptic Carmichael numbers introduced by Silverman to analogues of Euler-Jacobi and strong pseudoprimes. We investigate the relationships among Euler Elliptic Carmichael numbers , strong elliptic Carmichael numbers, products of anomalous primes and elliptic Korselt numbers of Type I: The former two of these are introduced in this paper, and the latter two of these were introduced by Mazur (1973) and Silverman (2012) respectively. In particular, we expand upon a previous work of Babinkostova et al. by proving a conjecture about the density of certain elliptic Korselt numbers of Type I that are products of anomalous primes.

Under sufficiently strong assumptions about the first prime in an arithmetic progression, we prove that the number of Carmichael numbers up to$X$is$\gg X^{1-R}$, where$R=(2+o(1))\log \log \log \log X/\text{log}\log \log X$. This is close to Pomerance’s conjectured density of$X^{1-R}$with$R=(1+o(1))\log \log \log X/\text{log}\log X$.