Carmichael numbers and the sieve
William D. Banks, Tristan Freiberg
本文通过筛法证明存在无限多个Carmichael数，其素因子形式为$p = 1 + a^2 + b^2$。
利用筛法，我们证明了存在无限多个Carmichael数，其所有素因子都具有形式$p = 1 + a^2 + b^2$，其中$a,b \in \mathbb{Z}$。
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 $a,b \in{\mathbb Z}$.
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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 Carmichael numbers is described, with its implementation up to $10^{24}$. Statistics relevant to the distribution of three-prime Carmichael numbers are given, with particular reference to the conjecture of Granville and Pomerance in [A.Granville and C.Pomerance, Two contradictory conjectures concerning Carmichael numbers, Math. Comp. 71 (2001), 883-908].
A composite positive integer $n$ is said to be a {\it weak Carmichael number} if $$ \sum_{\gcd(k,n)=1\atop 1\le k\le n-1}k^{n-1}\equiv φ(n) \pmod{n}. \leqno(1) $$ It is proved that a composite positive integer $n$ is a weak Carmichael number if and only if $p-1\mid n-1$ for every prime divisor $p$ of $n$. This together with Korselt's criterion yields the fact that every Carmichael number is also a weak Carmichael number. In this paper we mainly investigate arithmetic properties of weak Carmichael numbers. Motivated by the investigations of Carmichael numbers in the last hundred years, here we …
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 variation''. We present further statistics on the distribution of Carmichael numbers.
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 $L=2^αP^2$, where $k\in \mathbb{N}$ and $P$ is an odd prime number. There are eleven such Carmichael numbers.
We prove that for each odd number k, the sequence (k2^n+1)_{n\ge 1} contains only a finite number of Carmichael numbers. We also prove that k=27 is the smallest value for which such a sequence contains some Carmichael number.
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 variation''. We present further statistics on the distribution of Carmichael numbers.
In 1987, Dan Gordon defined an elliptic curve analogue to Carmichael numbers known as elliptic Carmichael numbers. In this paper, we prove that there are infinitely many elliptic Carmichael numbers. In doing so, we resolve in the affirmative the question of whether there exist infinitely square-free, composite integers $n$ such that for every prime $p$ that divides $n$, $p+1|n+1$.
We extend our previous computations to show that there are 1401644 Carmichael numbers up to $10^{18}$. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a ``large prime variation''. We present further statistics on the distribution of Carmichael numbers.
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 prime factor is greater than a determined lower bound. In addition, we conduct a Monte Carlo simulation as part of a probabilistic algorithm to detect if a given composite number is Carmichael. We modify this highly accurate algorithm with a deterministic primality test to create a novel, more efficient algorithm that differentiates between Carmichael numbers …
Carmichael Numbers with a Specified Number of Prime Factors
本文证明，对任意充分大的整数 $R$，都存在恰好含有 $R$ 个素因子的 Carmichael 数。
We give an upper bound for the number elliptic Carmichael numbers $n \le x$ that have recently been introduced by J. H. Silverman. We also discuss several possible ways for further improvements.
Carmichael Numbers with Prime Numbers of Prime Factors
本文在Heath-Brown猜想下，证明了存在无穷多个Carmichael数，其素因子个数为素数。
We propose a generalization of Carmichael numbers, where the multiplicative group $\mathbb G_\mathrm{m} = \mathrm{GL}(1)$ is replaced by $\mathrm{GL}(m)$ for $m\geq 2$. We prove basic properties of these families of numbers and give some examples.
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 $M$
Improving on some recent results of Matomäki 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 residue class.
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 product of three distinct prime numbers.
We show that universal elliptic Carmichael numbers do not exist, answering a question of Silverman. Moreover, we show that the probability that an integer $n$, which is not a prime power, is an elliptic Carmichael number for a random curve $E$ with good reduction modulo $n$, is bounded above by $\mathcal{O}(\log^{-1} n)$. If we choose both $n$ and $E$ at random, the probability that $n$ is $E$-carmichael is bounded above by $\mathcal{O}(n^{-1/8+ε})$.
In this paper, we prove that there are infinitely many $n$ for which $rad(φ(n))|n-1$ but $n$ is not a Carmichael number. Additionally, we prove that for any $k\geq 3$, there exist infinitely many $n$ such that $φ(n)|(n-1)^k$ but $φ(n)\nmid (n-1)^{k-1}$. The constructs that we consider here are generalizations of Carmichael and Lehmer numbers, respectively, that were first formulated by Grau and Oller-Marcén.
Lehmer's totient problem asks if there exist composite integers n satisfying the condition phi(n)|(n-1), (where phi is the Euler-phi function) while Carmichael numbers satisfy the weaker condition lambda(n)|(n-1) (where lambda is the Carmichael universal exponent function). We weaken the condition further, looking at those composite n where each prime divisor of phi(n) also divides n-1. (So rad(phi(n))|(n-1).) While these numbers appear to be far more numerous than the Carmichael numbers, we show that their distribution has the same rough upper bound as that of the Carmichael numbers, a bound …
Advances in Tabulating Carmichael Numbers
本文通过按P大小分叉的算法大幅推进了卡迈克尔数表计算，找到49679870个小于10^{22}的卡迈克尔数，并给出更快的渐近方法与代价估计。
Carmichael Numbers in All Possible Arithmetic Progressions
本文证明每个算术级数要么含无穷多个卡迈克尔数要么没有，并给出判定准则，进而解决Alford等人关于欧拉函数比值的极限问题。
Let E/Q be an elliptic curve, let L(E,s)=\sum a_n/n^s be the L-series of E/Q, and let P be a point in E(Q). An integer n > 2 having at least two distinct prime factors will be be called an elliptic pseudoprime for (E,P) if E has good reduction at all primes dividing n and (n+1-a_n)P = 0 (mod n). Then n is an elliptic Carmichael number for E if n is an elliptic pseudoprime for every P in E(Z/nZ). In this note we describe two elliptic analogues of Korselt's criterion for Carmichael numbers, and we analyze elliptic Carmichael numbers of the form pq.
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 new algorithms for the subset product problem that exploit the non-uniform distribution of primes p with the property that p-1 divides a highly composite Λ.
Erdős 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 $x$. At the time, his data tables up to $25 \cdot 10^{9}$ appeared to support his conjecture. However, Pinch extended this data and showed that up to $10^{21}$, Pomerance's conjecture did not appear well-supported. Thus, the purpose of this paper is two-fold. First, we build upon the work of Pomerance and others to present an alternate conjecture regarding the di…
Bertrand's Postulate for Carmichael Numbers
本文证明了Carmichael数在特定区间内的存在性，回答了Alford等人提出的关于Bertrand假设类比的问题，并给出了更强的计数下界。
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.
We prove some new lower bounds for the counting function $\mathcal N_{\mathcal C}(x)$ of the set of Novák-Carmichael numbers. Our estimates depend on the bounds for the number of shifted primes without large prime factors. In particular, we prove that $\mathcal N_{\mathcal C}(x) \gg x^{0.7039-o(1)}$ unconditionally and that $\mathcal N_{\mathcal C}(x) \gg xe^{-(7+o(1))(\log x)\frac{\log\log\log x}{\log\log x}}$, under some reasonable hypothesis.
On Carmichael numbers of the form $2^np^m+1$
本文证明当固定奇数 $m\ge 5$ 时，形如 $2^np^m+1$ 的卡迈克尔数仅有有限多个，其中 $n$ 为正整数，$p$ 为素数。
On primary Carmichael numbers
本研究围绕本原卡迈克尔数展开，证明三因子卡迈克尔数构造多项式$U_3(t)$的特殊分解性质，明确其成为本原卡迈克尔数的条件，还探讨了与的士数、多边形数的关联。
Algorithms for Carmichael numbers
本文研究了寻找小于指定界限的 Carmichael 数的算法复杂性，提出了一种改进的表格算法并实现了更高效的计算方法。
Ours go to 211: Euler pseudoprimes to 47 prime bases (from Carmichael numbers)
本文研究了卡迈克尔数中包含的欧拉伪素数，并提出了一种快速算法来生成新的欧拉伪素数，其中某些伪素数能够通过前47个素数基的测试。
Carmichael numbers and least common multiples of $p-1$
本文研究固定 $GCD[p_i-1]$ 的 Carmichael 数计数，在等差数列首素数强猜想下证明任意偶数 $\nu$ 对应数量的下界，突破标准构造中 $K$ 随 $n$ 增长的限制。
On primes in special sequences with applications to Carmichael numbers
本文利用等差数列中与Λ(n)相关的指数和，改进了非齐次Beatty序列和Piatetski-Shapiro序列上素数分布的结果，并将其应用于改进卡迈克尔数相关结论。
Idempotence in Carmichael and Giuga numbers and its application to the Agoh-Giuga conjecture
本文通过定义算术函数刻画Giuga数，推广Carmichael数至特定Fermat伪素数，并基于幂等性给出Agoh-Giuga猜想反例的新必要条件，推测无整数满足。
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 whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdos for the usual Carmichael numbers) that indicates that for every m there should be infinitely many Carmichael numbers of order m. The argument suggests a method for finding examples of higher-order Carmichael numbers; we use the method to provide e…
On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-$p$ digits
该研究在$p$进数理论框架下给出Carmichael数的新刻画，揭示其与Bernoulli多项式分母、基$p$数字和的关联，定义了本原Carmichael数，证明Carmichael数可表为显式多边形数，并给出相关集合的模子集覆盖。
$(a,a)$-Carmichael numbers and greatest common divisors of $p-a$
研究(a,a)-卡迈克尔数，通过定义K及C_ν(X,a)，在关于算术级数中首个素数的强猜想下，证明对特定整数a和自然数ν，给出C_ν(X,a)的下界。
The Carmichael lambda function $λ(n)$ is defined to be the smallest positive integer $m$ such that $a^m \equiv 1 \pmod{n}$ for all $(a,n)=1.$ $λ_k(n)$ is defined to be the $k$th iterate of $λ(n).$ Let L(n) be the smallest $k$ for which $λ_k(n)=1.$ It's easy to show that $L(n) \ll \log n.$ It's conjectured that $L(n)\asymp \log\log n,$ but previously it was not known to be $o(\log n)$ for almost all $n.$ We will show that $L(n) \ll (\log n)^δ$ for almost all $n,$ for some $δ<1.$ We will also show $L(n) \gg \log\log n$ for almost all $n$ and conjecture a normal order for L(n).
In this work we consider the congruence $\sum_{j=1}^{n-1} j^{k(n-1)} \equiv -1 \pmod n$ for each $k \in \mathbb{N}$, thus extending Giuga's ideas for $k=1$. In particular, it is proved that a pair $(n,k)\in \mathbb{N}^2$ satisfies this congruence if and only if $n$ is prime or a Giuga Number and $λ(n) \mid k(n-1)$. In passing, we establish new characterizations of Giuga numbers and we study some properties of the numbers $n$ satisfying $λ(n) \mid k(n-1)$.
Lehmer's totient problem consists of determining the set of positive integers $n$ such that $φ(n)|n-1$ where $φ$ is Euler's totient function. In this paper we introduce the concept of $k$-Lehmer number. A $k$-Lehmer number is a composite number such that $φ(n)|(n-1)^k$. The relation between $k$-Lehmer numbers and Carmichael numbers leads to a new characterization of Carmichael numbers and to some conjectures related to the distribution of Carmichael numbers which are also $k$-Lehmer numbers.
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 variant of the Solovay-Strassen test. We explore the growth of the test's "Carmichael" numbers, giving some empirical results and a discussion of one particularly strong pattern which appears in the results.
The Prime Wavelet Tree: Compressing the Carmichael Numbers
本文提出局部与全局两种数论压缩方法，利用素数小波树压缩满足类Korselt条件的数字集，将小于10^24的3.08亿个卡迈克尔数从18.4GB文本压缩至588.7MB，位宽由80位降至约15.3位。
Iteration of the modular l-th power function f(x) = x^l (mod n) provides a common pseudorandom number generator (known as the Blum-Blum-Shub generator when l=2). The period of this pseudorandom number generator is closely related to λ(λ(n)), where λ(n) denotes Carmichael's function, namely the maximal multiplicative order of any integer modulo n. In this paper, we show that for almost all n, the size of λ(λ(n)) is n/exp((1+o(1))(log log n)^2 log log log n). We conjecture an analogous formula for the k-th iterate of λ. We deduce that for almost all n, the psuedorandom number generator described…
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 $R$ as a Carmichael ideal if $R/I$ is a Carmichael ring, and a Carmichael element of $R$ means it generates a Carmichael ideal. In this paper, we determine the structure of Carmichael rings and prove a generalization of Korselt's criterion for Carmichael ideals in Dedekind domains. We also study Carmichael elements of polynomial rings over finite fields (called Carmichael polynomials) by generalizing various classical results. For example, we…
The structure of the group $(\mathbb{Z}/n\mathbb{Z})^\star$ and Fermat's little theorem are the basis for some of best-known primality testing algorithms. Many related concepts arise: Euler's totient function and Carmichael's lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer's totient problem, Giuga's conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider the underlying group $\mathcal{G}_n:=\{a+bi\in\mathbb{Z}[i]/n\mathbb{Z}[i] : a^2+b^2\equiv 1\ \textrm{$\pmod n$}\}$. In particular we characterize Gau…
Given an integer $k$, define $C_k$ as the set of integers $n > \max(k,0)$ such that $a^{n-k+1} \equiv a \pmod{n}$ holds for all integers $a$. We establish various multiplicative properties of the elements in $C_k$ and give a sufficient condition for the infinitude of $C_k$. Moreover, we prove that there are finitely many elements in $C_k$ with one and two prime factors if and only if $k>0$ and $k$ is prime. In addition, if all but two prime factors of $n \in C_k$ are fixed, then there are finitely many elements in $C_k$, excluding certain infinite families of $n$. We also give conjectures abou…
Quantum computers can solve many number theory problems efficiently. Using the efficient quantum algorithm for order finding as an oracle, this paper presents an algorithm that computes the Carmichael function for any integer $N$ with a probability as close to 1 as desired. The algorithm requires $O((\log n )^3n^3)$ quantum operations, or $O(\log\log n (\log n)^4 n^2)$ operations using fast multiplication. Verification, quantum optimizations and applications to RSA and primality tests are also discussed.
Carmichael numbers of the form $2^n p^m+1$ with odd exponent $m$: refinements of a theorem of Florian Luca, a general independence lemma, and a computational study
本文推广了Luca关于形如$2^n p^m+1$（$m$为奇数）的Carmichael数的定理，给出显式界、推广至任意偶数底数，并通过计算验证了$10^{18}$内唯一例子为1729。
We suggest other models of sieve generated sequences like the Sieve of Eratosthenes to explain randomness properties of the prime numbers, like the twin prime conjecture, the lim sup conjecture, the Riemann conjecture, and the prime number theorem.
We establish large sieve inequalities for power moduli in imaginary quadratic number fields, extending earlier work of Baier and Bansal for the Gaussian field.
We provide elementary and accurate numerical solutions to the differential-difference equation, which improves an explicit version of the linear sieve given by Nathanson.
We make two algorithms that generate all prime numbers up to a given limit, they are a development of sieve of Eratosthenes algorithm, we use two formulas to achieve this development, where all the multiples of prime number 2 are eliminated in the first formula, and all the multiples of prime numbers 2 and 3 are eliminated in the second formula. Using the first algorithm we proof sieve of Sundaram's algorithm, then we improve it to be more efficient prime generating algorithm. We will show the difference in performance between all the algorithms we will make and sieve of Eratosthenes algorithm…
On a certain arithmetic function defined via Bernoulli numbers
研究定义在奇数\(n\geq3\)上、通过伯努利数构造的算术函数，该函数将素数、卡迈克尔数和久加数统一到一个整除性准则中，为相关研究提供新视角。
We formulate and prove a large sieve inequality for quadratic characters over a number field. To do this, we introduce the notion of an n-th order Hecke family. We develop the basic theory of these Hecke families, including versions of the Poisson summation formula.
Factorisation of integers $n$ is of number theoretic and cryptographic significance. The Number Field Sieve (NFS) introduced circa 1990, is still the state of the art algorithm, but no rigorous proof that it halts or generates relationships is known. We propose and analyse an explicitly randomised variant. For each $n$, we show that these randomised variants of the NFS and Coppersmith's multiple polynomial sieve find congruences of squares in expected times matching the best-known heuristic estimates.
In this thesis we give an in-depth introduction to the General Number Field Sieve, as it was used by Buhler, Lenstra, and Pomerance, before looking at one of the modern developments of this algorithm: A randomized version with provable complexity. This version was posited in 2017 by Lee and Venkatesan and will be preceded by ample material from both algebraic and analytic number theory, Galois theory, and probability theory.
Explicit sieve estimates and nonexistence of odd multiperfect numbers of a certain form
本文通过证明筛法函数的显式渐近公式，证明了不存在丰度为四且平方部分无立方因子的奇多重完全数。
Nonlinear polynomial selection algorithms for the number field sieve address the problem of constructing polynomials with small coefficients by reducing to instances of the well-studied problem of finding short vectors in lattices. The reduction rests upon the construction of modular geometric progressions with small terms. In this paper, the methods used to construct the geometric progressions are extended, resulting in the development of two nonlinear polynomial selection algorithms.
On Types of Elliptic Pseudoprimes
本文推广椭圆伪素数概念，定义Euler椭圆Carmichael数与强椭圆Carmichael数，研究其与异常素数乘积及Type I椭圆Korselt数的关系，并证明相关密度猜想。
We derive explicit formulas for integrals of certain symmetric polynomials used in Keiju Sono's multidimensional sieve of $E_2$-numbers, i.e., integers which are products of two distinct primes. We use these computations to produce the currently best-known bounds for gaps between multiple $E_2$-numbers. For example, we show there are infinitely many occurrences of four $E_2$-numbers within a gap size of 94 unconditionally and within a gap size of 32 assuming the Elliott-Halberstam conjecture for primes and sifted $E_2$-numbers.
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