On the greatest prime factor of (ab+1)(ac+1)

多项式值的最大素因子与族中的次指数Szpiro猜想

On the greatest prime factor of polynomial values and subexponential Szpiro in families

本文结合模方法与对数线性形式，为椭圆曲面上的Szpiro猜想及二次、三次多项式值的最大素因子下界问题获得改进的无条件界，并扩展了先前关于$n^2+1$的结果。

例外Maass形式的大型筛不等式及n²+1的最大大素因子

Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$

本文通过证明例外Maass形式的新型大型筛不等式，改进了n²+1的最大大素因子的下限，并应用于素数和光滑数在算术级数中的分布指数。

Greatest common divisors of shifted primes and Fibonacci numbers

Let $(F_n)$ be the sequence of Fibonacci numbers and, for each positive integer $k$, let $\mathcal{P}_k$ be the set of primes $p$ such that $\gcd(p - 1, F_{p - 1}) = k$. We prove that the relative density $\text{r}(\mathcal{P}_k)$ of $\mathcal{P}_k$ exists, and we give a formula for $\text{r}(\mathcal{P}_k)$ in terms of an absolutely convergent series. Furthermore, we give an effective criterion to establish if a given $k$ satisfies $\text{r}(\mathcal{P}_k) > 0$, and we provide upper and lower bounds for the counting function of the set of such $k$'s. As an application of our results, we give …

On prime factors of terms of binary recurrence sequences

We give estimates from below for the greatest prime factor of the n-th term of a binary recurrence sequence.

关于具有小素因子的整数序列

On sequences of integers with small prime factors

本文证明了最大素因子缓慢增长的整数序列中，相邻项的差趋于无穷大。

Arithmetic properties of blocks of consecutive integers

This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of consecutive integers, both without any assumption and under assumption of the abc-conjecture. Finally we prove that the explicit abc-conjecture implies the Erdős-Woods conjecture for each k>2.

计算二项式系数 $\binom{mn}{mk}$ 的最大公约数

Computing the Greatest Common Divisor of Binomial Coefficients $\binom{mn}{mk}$

本文研究了二项式系数 $\binom{mn}{mk}$ 的最大公约数，给出了 $m=2$ 时该 GCD 的完整刻画，并展示了在特定条件下高效计算的方法。

无穷大素数模下的Tonelli–Shanks幂的超越性

Transcendence of Tonelli--Shanks Power Modulo Infinitely Large Primes

该数学研究证明了无穷大素数模整数环中，与Tonelli–Shanks算法相关的多个数论对象的像在有理数域上具有超越性。

关于多项式整数部分的最大公约数

On the greatest common divisor of integer parts of polynomials

本文受Bergelson和Richter的问题启发，研究了正整数与多项式整数部分构成互素元组的计数问题，并给出了误差项强度依赖于首项系数丢番图性质的渐近公式。

On consecutive quadratic non-residues: a conjecture of Issai Schur

Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest number of consecutive quadratic non-residues modulo p is always less than the square root of p. This paper uses elementary methods to prove that 13 is the only prime number for which the greatest number of consecutive quadratic non-residues modulo p exceeds the square root of p.

Quadratic polynomials at prime arguments

For a fixed quadratic irreducible polynomial $f$ with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes $p$ such that $f(p)$ has at most 4 prime factors, improving a classical result of Richert who requires 5 in place of 4. Denoting by $P^+(n)$ the greatest prime factor of $n$, it is also proved that $P^+(f(p))>p^{0.847}$ infinitely often.

Integers for Radical Extensions of Odd Prime Degree as Product of Subrings

For a radical extension K of odd prime degree the ring O_K of integers is constructed as a product of subrings with the following property: for all prime divisors q of the discriminant of O_K there is a q-maximal factor. The discriminant of O_K is the greatest common divisor of the discriminants of all factors. The results are applied to give a criterion for the monogeneity of K where the opposite is not true.

On the greatest common divisor of $n$ and the $n$th Fibonacci number

Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. We prove that $\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x$ for all $x \geq 2$, and that $\mathcal{A}$ has zero asymptotic density. Our proofs rely on a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$.

Estimating the greatest common divisor of the value of two polynomials

Let $p$ be a fixed prime, and let $v(a)$ stand for the exponent of $p$ in the prime factorization of the integer $a$. Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Write $S$ for the maximum of $v(\gcd (f(n), g(n)))$ over all integers $n$. It is known that $S \le v(r)$. We give various lower and upper bounds for the least possible value of $v(r)-S$ provided that a given power $p^s$ divides both $f(n)$ and $g(n)$ for all $n$. In particular, the least possible value is $ps^2-s$ for $s\le p$ and is asymptotically $(p-1)s^2$ for large $s$.

(a,a)-卡迈克尔数与p - a的最大公因数

$(a,a)$-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。

同时等差数列中的素数

Primes in simultaneous arithmetic progressions

本文通过结合谱理论和代数几何方法，证明了两个同时等差数列中素数分布的新均值定理，并展示了Chen素数p + 6的最大素因子的界

floor(n^α)和floor(n^β)同时为质数的频率如何？

How often are $ \lfloor {n^α} \rfloor $ and $ \lfloor {n^β} \rfloor $ simultaneously primes?

该研究证明存在无限多个正整数n，使得多个floor(n^α)同时为质数，并基于等分布定理提出方法。

莫比乌斯函数的一个令人惊讶的推广

A surprising generalization of the Möbius function

本文推广了莫比乌斯函数的递归公式，用任意算术函数替换最大整数函数，定义新函数族，并证明其值为-1当且仅当参数为素数，从而给出素数的代数刻画。

关于完美幂与整数S-单位的差异

On the difference between perfect powers and integral $S$-units

研究通过建立有效下界，探讨了正整数z与整数S-单位之间的差异，揭示了其最大素因子随z^d增长的趋势。

关于平方数与积分 $S$-单位之间的差异

On the difference between squares and integral $S$-units

本文针对平方数与积分 $S$-单位的差，建立了其最大素因子的有效下界，该下界随相关参数趋于无穷大。

Carmichael 数与素数个数的素因子

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。

On shifted primes with large prime factors and their products

We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set of positive integers n up to x with exactly k prime factors, counted with or without multiplicity, such that the largest prime factor of gcd(p-1 : p | n) exceeds n^{1/2k}.

Product of an integer free of small prime factors and prime in arithmetic progression

We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. Our best results is conditional on the Generalised Riemann hypothesis (GRH). As a corollary, we make progress on a conjecture of Erdös, Odlyzko and Sárközy.

All unitary perfect polynomials over $F_2$ with less than five distinct prime factors

We find all unitary perfect polynomials over the prime field $\F_2$ with less than five distinct prime factors.

An iteration based on prime and composite factors

A discrete map based on the sum of an integer's distinct primes factors and the sum of its other factors is defined and its iteration is studied.

On the Sum and Product of Distinct Prime Factors of an Odd Perfect Number

We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.

On a conjecture on shifted primes with large prime factors

Let $\mathcal{P}$ be the set of all primes and $π(x)$ be the number of primes up to $x$. For any $n\ge 2$, let $P^+(n)$ be the largest prime factor of $n$. For $0<c<1$, let $$T_c(x)=\#\{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\}.$$ In this note, we proved that there exists some $c<1$ such that $$\limsup_{x\rightarrow\infty}\frac{T_c(x)}{π(x)}<\frac12,$$ which disproves a conjecture of Chen and Chen.

Novák-Carmichael numbers and shifted primes without large prime factors

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.

平均Brun-Titchmarsh定理与具有大素因子的移位素数

An average Brun-Titchmarsh theorem and shifted primes with a large prime factor

该研究利用Maynard的三重卷积估计突破，改进了平均Brun-Titchmarsh定理，并应用于移位素数，证明无穷多个素数$p$满足$p-1$的最大素因子大于$p^{0.679}$。

Odd Perfect Numbers Have At Least Nine Distinct Prime Factors

An odd perfect number, N, is shown to have at least nine distinct prime factors. If 3 does not divide N, then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.

A binomial identity on the least prime factor of an integer

An identity for binomial symbols modulo an odd positive integer $n$ relating to the least prime factor of $n$ is proved. The identity is discussed within the context of Pell conics.

Primes in arithmetic progressions to large moduli, and shifted primes without large prime factors

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.

On the digital representation of integers with bounded prime factors

Let $b \ge 2$ be an integer. Not much is known on the representation in base $b$ of prime numbers or of numbers whose prime factors belong to a given, finite set. Among other results, we establish that any sufficiently large integer which is not a multiple of $b$ and has only small (in a suitable sense) prime factors has at least four nonzero digits in its representation in base $b$.

$\operatorname{SL}_2(\mathbb{R})$ 自守核的加权平均，第一部分：非振荡函数

Weighted averages of $\operatorname{SL}_2(\mathbb{R})$ automorphic kernel, Part I: non-oscillatory functions

本文证明了一个计算由 $\operatorname{SL}_2(\mathbb{Z})$ 同余子群参数化的和的加权平均的定理，通过将谱方法直接应用于自守核以更好地保留对称性，并应用于二次多项式获得了关于最大素因子和根等分布的新结果。

关于小素因子数量的局部规律

On local laws for the number of small prime factors

本文研究Alladi提出的整数小素因子局部分布问题，重点刻画参数特定取值下出现的过渡阶段现象。

具有指定素因子数量的 Carmichael 数

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。

On the prime factors of the iterates of the Ramanujan $τ$--function

In this paper, for a positive integer $n\ge 1$, we look at the size and prime factors of the iterates of the Ramanujan $τ$ function applied to $n$.

On non-congruent numbers with 3 modulo 8 prime factors

In this paper, we will give a series of non-congruent numbers with $\equiv3\pmod8$ prime factors.

Explicit Factorization of Prime Integers in Quartic Number Fields defined by $X^4+aX+b$

For every prime integer $p$, an explicit factorization of the principal ideal $p\z_K$ into prime ideals of $\z_K$ is given, where $K$ is a quartic number field defined by an irreducible polynomial $X^4+aX+b\in\z[X]$.

Signed harmonic sums of integers with $k$ distinct prime factors

We give some theoretical and computational results on "random" harmonic sums with prime numbers, and more generally, for integers with a fixed number of prime factors.

Beurling numbers whose number of prime factors lies in a specified residue class

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.

The largest prime factor of $X^3+2$

Improving on a theorem of Heath-Brown, we show that if $X$ is sufficiently large then a positive proportion of the values $\{n^3+2:n\in (X,2X]\}$ have a prime factor larger than $X^{1+10^{-52}}$.

On a problem of Erdős involving the largest prime factor of $n$

An asymptotic formula for the number of $n \le x$ such that $n$ does not divide $P(n)!$ is given, where P(n) is the largest prime factor of $n$.

Biases in prime factorizations and Liouville functions for arithmetic progressions

We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arithmetic progression in the prime factorizations of integers. For example, we observe that the primes of the form $4k+1$ tend to appear an even number of times in the prime factorization of a given integer, more so than for primes of the form $4k+3$. We are led to consider variants of Pólya's conjecture, supported by extensive numerical evidence, and its relation to other conjectures.

素数元组猜想下素因子级数的无理性

The irrationality of a prime factor series under a prime tuples conjecture

本文在素数 $k$-tuples 猜想下证明了级数 $\sum_{n=1}^\infty \frac{ω(n)}{2^n}$ 是无理数，有条件地解决了 Erdős 提出的一个问题。

On the largest prime factor of $x^2-1$

In this paper, we find all integers $x$ such that $x^{2}-1$ has only prime factors smaller than 100. This gives some interesting numerical corollaries. For example, for any positive integer $n$ we can find the largest positive integer $x$ such that all prime factors of each of $x, x+1,..., x+n$ are less than 100.

On inequalities involving counts of the prime factors of an odd perfect number

Let $N$ be an odd perfect number. Let $ω(N)$ be the number of distinct prime factors of $N$ and let $Ω(N)$ be the total number (counting multiplicity) of prime factors of $N$. We prove that $\frac{99}{37}ω(N) - \frac{187}{37} \leq Ω(N)$ and that if $3\nmid N$, then $\frac{51}{19}ω(N)-\frac{46}{19} \leq Ω(N)$.

Even perfect polynomials over $F_2$ with four prime factors

A perfect polynomial over the binary field $\F_2$ is a polynomial $A \in \F_2[x]$ that equals the sum of all its divisors. If $\gcd(A,x^2-x) \neq 1$ then we call $A$ even. The list of all even perfect polynomials over $\F_2$ with at most 3 prime factors in known. The object of this paper is to give the list of all even perfect polynomials over $\F_2$ with four prime factors. These are all the known perfect polynomials with four prime factors over $\F_2$.

Reducing the number of prime factors of long $κ$-tuples

We prove that there are infinitely many integers $n$ such that the total number of prime factors of $(n+h_{1})(n+h_{2})...(n+h_κ)$ is at most $(1/2)κ\logκ+O(κ)$, provided $κ$ is sufficiently large.

3-tuples have at most 7 prime factors infinitely often

Let $L_1$, $L_2$ $L_3$ be integer linear functions with no fixed prime divisor. We show there are infinitely many $n$ for which the product $L_1(n)L_2(n)L_3(n)$ has at most 7 prime factors, improving a result of Porter. We do this by means of a weighted sieve based upon the Diamond-Halberstam-Richert multidimensional sieve.

Primitive prime factors in second order linear recurrence sequences

For a class of Lucas sequences ${x_n}$, we show that if $n$ is a positive integer then $x_n$ has a primitive prime factor which divides $x_n$ to an odd power, except perhaps when $n = 1, 2, 3 or 6$. This has several desirable consequences.

Summatory function of the number of prime factors

We consider the summatory function of the number of prime factors for integers $\leq x$ over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg Martin conjectured that the difference of the summatory functions should attain a constant sign for all sufficiently large $x$. In this paper, we provide strong evidence for Greg Martin's conjecture. Moreover, we derive a general theorem for arithmetic functions from the Selberg class.

关于一个素数和一个与任何整数互质且至多有两个素因子的平方数之和

On the sum of a prime and a square-free number co-prime to any integer with at most two prime factors

研究证明大于2的自然数可表示为素数与特定平方数之和，并明确给出了渐进行为结果。

Joint Poisson distribution of prime factors in sets

Given disjoint subsets $T_1,\ldots,T_m$ of "not too large" primes up to $x$, we establish that for a random integer $n$ drawn from $[1,x]$, the $m$-dimensional vector enumerating the number of prime factors of $n$ from $T_1,\ldots,T_m$ converges to a vector of $m$ independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when $T_1,\ldots,T_m$ are unrestricted, and apply this to the distribution of the number of prime factors from a set $T$ given that $n$ has $k$ total prime factor…