Carmichael Numbers with a Specified Number of Prime Factors
Daniel Larsen, Thomas Wright
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。
对于每个充分大的整数 $R$，都存在一个恰有 $R$ 个素因子的 Carmichael 数。
For every sufficiently large integer $R$, there exists a Carmichael number with exactly $R$ prime factors.
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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].
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 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.
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 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.
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.
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.
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。
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.
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。
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.
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 …
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 …
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.
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 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。
Carmichael numbers and the sieve
本文通过筛法证明存在无限多个Carmichael数，其素因子形式为$p = 1 + a^2 + b^2$。
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。
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 $\mathcal{O}(n^{-1/8+ε})$.
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。
A new characterization for the Lucas-Carmichael Integers and sums of base-$p$ digits
本文通过基-$p$ 位数和给出 Lucas-Carmichael 整数的充分必要刻画，研究其相关性质，并在素数 $k$-元组猜想下证明其无穷性。
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 …
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.
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 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。
Runs of integers with constant values of the Carmichael function
本文研究Carmichael λ函数在连续整数上取相同值的最大连续段长度，通过引入基于Baker-Harman关于移位素数最大素因子结果的反向计数论证，改进了2023年已有的上界估计。
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…
$(a,a)$-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。
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 $λ(n) \mid k(n-1)$.
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。
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。
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。
In this work, we obtain some new lower bounds for the number $\mathcal N_B(x)$ of Novák numbers less than or equal to $x$. We also prove, conditionally on Generalized Riemann Hypothesis, the upper estimates for the number of primes dividing at least one Novák number and give description for the prime factors of Novák numbers $N$, such that $2N$ is a Novák-Carmichael number.
We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the the distribution of Carmichael numbers. Our main technical result is a new mean value theorem for primes in arithmetic progressions to large moduli. Namely, we estimate primes of size $x$ with quadrilinear forms of moduli up to $x^{17/32}$. This extends moduli beyond $x^{11/21}$, recently obtained by Maynard, improving $x^{29/56}$ from well-known 1986 work of Bombieri, Friedlander, and Iwaniec.
Tabulating Absolute Lucas Pseudoprimes
研究固定判别式下绝对 Lucas 伪素数的枚举问题，提出受 Carmichael 数启发的枚举算法及渐近分析，证明给定素因子集合下伪素数有限，并首次给出判别式 5 的枚举结果。
Carmichael showed for sufficiently large $L$, that $F_L$ has at least one prime divisor that is $\pm 1({\rm mod}\, L)$. For a given $F_L$, we will show that a product of distinct odd prime divisors with that congruence condition is a Fibonacci pseudoprime. Such pseudoprimes can be used in an attempt, here unsuccessful, to find an example of a Baillie-PSW pseudoprime, i.e.\ an odd Fibonacci pseudoprime that is congruent to $\pm 2({\rm mod}\, 5)$ and is also a base-2 pseudoprime.
Under sufficiently strong assumptions about the first term in an arithmetic progression, we prove that for any integer $a$, there are infinitely many $n\in \mathbb N$ such that for each prime factor $p|n$, we have $p-a|n-a$. This can be seen as a generalization of Carmichael numbers, which are integers $n$ such that $p-1|n-1$ for every $p|n$.
Suppose that $α,β\in\mathbb{R}$. Let $α\geqslant1$ and $c$ be a real number in the range $1<c< 12/11$. In this paper, it is proved that there exist infinitely many primes in the generalized Piatetski--Shapiro sequence, which is defined by $(\lfloorαn^c+β\rfloor)_{n=1}^\infty$. Moreover, we also prove that there exist infinitely many Carmichael numbers composed entirely of primes from the generalized Piatetski--Shapiro sequences with $c\in(1,\frac{19137}{18746})$. The two theorems constitute improvements upon previous results by Guo and Qi.
In this work we present a natural generalization of strong pseudoprime to base $b$, which we have called $q$-pseudoprime to base $b$. It allows us to present another way to define a Midy's number to base $b$ (overpseudoprime to base $b$). Besides, we count the bases $b$ such that $N$ is a $q$-probable prime base $b$ and those ones such that $N$ is a Midy's number to base $b$. Furthemore, we prove that there is not a concept analogous to Carmichael numbers to $q$-probable prime to base $b$ as with the concept of strong pseudoprimes to base $b$.
We consider various arithmetic questions for the Piatetski-Shapiro sequences $\fl{n^c}$ ($n=1,2,3,...$) with $c>1$, $c\not\in\N$. We exhibit a positive function $θ(c)$ with the property that the largest prime factor of $\fl{n^c}$ exceeds $n^{θ(c)-\eps}$ infinitely often. For $c\in(1,\tfrac{149}{87})$ we show that the counting function of natural numbers $n\le x$ for which $\fl{n^c}$ is squarefree satisfies the expected asymptotic formula. For $c\in(1,\tfrac{147}{145})$ we show that there are infinitely many Carmichael numbers composed entirely of primes of the form $p=\fl{n^c}$.
We study the question of whether for each n there is another integer m with lambda(m)=lambda(n), where lambda is Carmichael's function. We give a "near" proof of the fact that this is the case unconditionally, and a complete conditional proof under the Extended Riemann Hypothesis. The main tool is a count of prime chains from paper arXiv:0906.3380.
We find asymptotics for $S_{K,c}(x)$, the number of positive integers below $x$ whose number of prime factors is $c \; \mathrm{mod}\; K$. We study this question in the context of Beurling integers.
T-Fermat integers
引入T-费马整数概念，推广素数定义；证明合数T-费马整数必为卡迈克尔数，并猜想存在无穷多个合数T-费马整数，为证明ω(p-1)≤2的素数无穷提供可能路径。
We extend known results on the number of solutions to a linear equation in at least three prime numbers when the primes involved are required to lie in specified Chebotarev classes. We prove asymptotic results similar to previous ones only now taking into account corrections coming form the Chebotarev Density Theorem and Global Class Field Theory. We then apply these results to find elliptic curves whose discriminants split completely of a given number field.
Imaginary quadratic fields with $\ell$-torsion-free class groups and specified split primes
本文证明存在类数不被给定奇素数 ℓ 整除、且在指定奇素数集合 S_+ 中完全分裂的虚二次域，通过改进 Hurwitz 类数非全纯 Ramanujan 型同余分类，提出了不同于 Wiles 结果的新方法，且不要求分裂素数满足 p ≢ -1 (mod ℓ)。
Definition of the invariant and the relationship with the compounds numbers. Generalisation of the Euler theorem
本文引入不变量的概念及其性质用于素性检验，结合欧拉定理将其推广至所有 $a^{φ(m)}$ 情形，并提出替代 Totient 函数以解释 Carmichael 数的新假设。
We revisit the problem of integer factorization with number-theoretic oracles, including a well-known problem: can we factor an integer $N$ unconditionally, in deterministic polynomial time, given the value of the Euler totient $$Φ$(N)$? We show that this can be done, under certain size conditions on the prime factors of N. The key technique is lattice basis reduction using the LLL algorithm. Among our results, we show for example that if $N$ is a squarefree integer with a prime factor $p > $\sqrt$ N$ , then we can recover p in deterministic polynomial time given $$Φ$(N)$. We also shed some li…
The Hardy--Ramanujan inequality for sifted sets and its applications
本文建立了适用于筛集的Hardy-Ramanujan不等式加权版本，并将其应用于多个数论问题，包括大偏差、乘法表问题和移位素数因数等。
We show by an inclusion-exclusion argument that the prime $k$-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect of a theorem of Gallagher that the prime $k$-tuple conjecture implies that the prime numbers are distributed in a Poisson distribution around their average spacing.
Let $A$ be a $g$-dimensional abelian variety over $\mathbb{Q}$ whose adelic Galois representation has open image in $\text{GSp}_{2g} \widehat{\mathbb{Z}}$. We investigate the endomorphism algebras $\text{End}(A_p) \otimes \mathbb{Q} = \mathbb{Q}( π_p )$ of the reduction of $A$ modulo primes $p$ at which this reduction is ordinary and simple. We obtain conditional and unconditional asymptotic upper bounds on the number of primes at which this "Frobenius field" is a specified number field and, when $A$ is two-dimensional, at which the Frobenius field contains a specified real quadratic number fi…
Probabilistic analytical approach to determining the asymptotics of prime objects on the initial interval of the natural series
本文提出一种基于概率空间的概率分析方法，用于确定自然数列初始区间上素数对象的渐近性，并验证其与已知素数猜想一致，为证明素数猜想提供新途径。
Notes on Mitsui's Prime Number Theorem with Siegel zeros
本文通过纳入Hecke L-函数的Siegel零点，改进了三井1957年的素数定理，使模的范数能以伪多项式速率增长，为后续研究素元素线性模式提供关键支持。
Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings
本文证明三素数整数平铺中存在无支撑分圆因子，刻画了平方周期情形，并给出最小模数及Apéry集构造。
The universal profile of the invariant factors of $({\mathbb Z}/n{\mathbb Z})^\times$
本文研究乘法群Mₙ=(ℤ/nℤ)⁺的不变因子结构，证明典型整数n的Mₙ除最大因子外的不变因子渐近具有通用分布，并建立了关于等于ℤ/kℤ的不变因子个数的Erdős-Kac型定理（分布为偏正态或相关分布）。
Iterated function systems (IFS) can be a surprisingly useful tool for studying structure in data. Here we present results stemming from a 2013 computational study by the author using IFS. The results include fractal patterns that reveal "repulsive" phenomena among primes in a wide range of classes, having specified arithmetic or congruence properties. Some of the phenomena shown in our computations relate to the recent, groundbreaking work of Lemke Oliver and Soundararajan on biases between consecutive primes. We do not have asymptotics to explain our results, but provide graphs, data, and det…
Let $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to construct explicitly elements of high order in the field $F_q[x]/\langle x^m-a\rangle $. Namely, we find elements with multiplicative order of at least $5^{\sqrt[3]{m/2}}$, which is better than previously obtained bound for such family of extension fields.
Linear patterns of prime elements in number fields
研究数字段中一次多项式同时取素数值的结构，应用包括有理点哈塞原理和椭圆曲线秩构造，推翻广义希策诺夫第十问题的肯定回答。
Exploring Prime Number Classification: Achieving High Recall Rate and Rapid Convergence with Sparse Encoding
本文提出一种结合机器学习与数论的新方法，通过稀疏编码与传统神经网络结合，实现高召回率和快速收敛，验证了在素数识别中的有效性。
A Proof of Bala's Congruence Conjectures for A158690
本文证明了Bala关于OEIS序列A158690的两个同余猜想，通过一般定理和Stirling数公式，得到模周期性与移位Gauss同余。
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