Aleksandar Ivić
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 $n$.
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In the present work we prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent fantastic achievement of Yitang Zhang about the existence of bounded gaps between consecutive primes. Most of these results would have belonged to the category of science fiction a decade ago. However, the presented results are far from being immediate consequences of Zhang's famous theorem: they require various new ideas, other important properties of the applied sieve functi…
The convergence of an alternating series of Erdős, assuming the Hardy--Littlewood prime tuples conjecture
本文针对Erdős提出的某交错级数收敛性的公开问题，结合素数随机筛选模型与Gallagher计算变体，在强版本Hardy–Littlewood素数元组猜想下证明了该级数条件收敛。
Let p_n denote the sequence of all primes and let d_n=p_n-p_{n-1} denote the sequence of all gaps between consecutive primes. In 1948 Erdős and Turán showed that d_{n+1}-d_n changes sign infinitely often and together with Pólya asked for a necessary and sufficient condition that a fixed linear combination of r>1 consecutive prime gaps d_{n+i} i=(1,2,...r) should change sign infinitely often as n runs through the sequence of all natural numbers. They conjectured that the condition is that the non zero values among the coefficients of the linear combination cannot all have the same sign but they…
We provide a way to modify and to extend a previously established inequality by P. Erdős, R. Graham and others and to answer a conjecture posed in the nineties by R. Graham, which bears on the lack of divisibility of the central binomial coefficient by three distinct, fixed odd primes. In fact the result will show by using an approach similar to their own which they proved for the case of two fixed odd primes, that the central binomial coefficient is not divisible infinitely often by three distinct and fixed odd primes. Therefore a generalization to more fixed odd primes than three but finite …
On the largest value of the solutions of Erdős's last equation
本文研究Erdős最后方程解的最大值，证明其随$n$趋于无穷而趋于无穷，并完整刻画了$x_n\le 10$的所有解。
In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume $E_2$-values; i.e., values that are products of exactly two primes. We use that result to prove that there are inifinitely many integers $x$ that simultaneously satisfy $$ω(x)=ω(x+1)=4, Ω(x)=Ω(x+1)=5, \text{and} d(x)=d(x+1)=24.$$ Here, $ω(x), Ω(x), d(x)$ represent the number of prime divisors of $x$, the number of prime power divisors of $x$, and the number of divisors of $x$, respectively. We also prove simila…
In the present work we prove a common generalization of Maynard-Tao's recent result about consecutive bounded gaps between primes and on the Erdős-Rankin bound about large gaps between consecutive primes. The work answers in a strong form a 60 years old problem of Erdös, which asked whether the ratio of two consecutive primegaps can be infinitely often arbitrarily small, and arbitrarily large, respectively.
A Complete Answer to Erdős Problem 690
本文基于埃德华问题690，证明对于k≥4，p↦d_k(p)序列非单峰，通过多项式比值与素数间隙比较，结合素数计数函数估计和构造，得出严格下降后上升的结论。
We introduce a variation on the prime divisor function $B(n)$ of Alladi and Erdős, a close relative of the sum of proper divisors function $s(n)$. After proving some basic properties regarding these functions, we study the dynamics of its iterates and discover behaviour that is reminiscent of aliquot sequences. We prove that no unbounded sequences occur, analogous to the Catalan-Dickson conjecture, and give evidence towards the analogue of the Erdős-Granville-Pomerance-Spiro conjecture on the pre-image of $s(n)$.
For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ with $x,y,z$ positive integers. The \emph{Erdős-Straus conjecture} asserts that $f(n) > 0$ for every $n \geq 2$. To solve this conjecture, it suffices without loss of generality to consider the case when $n$ is a prime $p$. In this paper we consider the question of bounding the sum $\sum_{p<N} f(p)$ asymptotically as $N \to \infty$, where $p$ ranges over primes. Our main result establishes the asymptotic upper and lower bounds $$ N \log^2 N…
Generalizations of the Erdős-Kac Theorem and the Prime Number Theorem
本文研究整数素数因子分布与最大素因子分布的线性无关性，改进了埃罗什-卡茨定理和洛伊德的结果，并展示了这些结果在埃罗什-波默兰茨定理下的类比性。
Erdös conjectured that the set J of limit points of d_n/logn contains all nonnegative numbers, where d_n denotes the nth primegap. The author proved a year ago (arXiv: 1305.6289) that J contains an interval of type [0,c] with a positive ineffective value c. In the present work we extend this result for a large class of normalizing functions. The only essential requirement is that the function f(n) replacing logn should satisfy f(n)<<lognloglognloglogloglogn/(logloglogn)^2 (with a small implied constant), the well-known Erdös-Rankin bound for the largest known gaps between consecutive primes. T…
Sparse Admissible Sets and a Problem of Erdős and Graham
本文通过基于原根幂的贪心构造，证明存在任意稀疏的无限容许集无法平移为素数子集，否定地回答了 Erdős 和 Graham 的问题，并补充了三种构造。
On the Erdős--Odlyzko--Sárközy conjecture for prime-power moduli
该研究针对素数幂模q=p^γ，证明Erdős–Odlyzko–Sárközy猜想中第三个素数可取次多项式小量，给出相关渐近公式并验证多数既约剩余类满足猜想，还提供了任意q下的条件版本。
On Erdős problem $\# 648$
本文确定了满足元素最大素因子递减且元素不超过 $n$ 的递增序列的最大长度阶，解决了 Erdős 问题 #648。
A subset of the integers larger than 1 is $primitive$ if no member divides another. Erdos proved in 1935 that the sum of $1/(a\log a)$ for $a$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained for the set of prime numbers. In this paper we make some progress on several fronts, and show a connection to certain prime number "races" such as the race between $π(x)$ and li$(x)$.
This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erdős-Mirksy conjecture which states that there are infinitely many positive integers $x$ such that $d(x)=d(x+1)$ where $d(x)$ denotes the number of divisors of $x$. This conjecture was first proven by Heath-Brown in 1984, but the method did not reveal the nature of the set of values $d(x)$ for such $x$. In particular, one could not conclude that there was any particular value $A$ for which $d(x)=d(x+1)=A$ infinitely often. In the previous paper …
Constructive Proofs of the Erdos-Straus Conjecture for Prime Numbers with P congruent to 1 modulo 4
本文针对 P ≡ 1 mod 4 的素数，提出两种构造性方法证明 Erdős–Straus 猜想，其中方法 ED2 将解集刻画为仿射格并显式构造表示，同时引入卷积与反卷积算法并通过大规模计算验证其正确性与有效性。
A solution to the Straus-Erdős conjecture
本文提出了解决斯特劳斯-埃罗德猜想的方法，通过找到满足特定等式的正整数x、y、z来证明猜想成立。
Non-Wieferich property of prime ideals and a conjecture of Erdös
本文研究了素理想在特定条件下的非魏尔费尔德性质，并推广了杜普-魏尔费尔德定理，探讨了β进制展开中数字的渐近等分布及块复杂度。
This paper studies the largest cycles consisted by the quadratic residues modulo prime numbers. We give some formulae about the maximum length of the cycles. Especially, the formula for modulo Fermat primes is given.
A set of natural numbers is primitive if no element of the set divides another. Erdős conjectured that if S is any primitive set, then \sum_{n\in S} 1/(n log n) \le \sum_{n\in ¶} 1/(p log p), where ¶denotes the set of primes. In this paper, we make progress towards this conjecture by restricting the setting to smaller sets of primes. Let P denote any subset of ¶, and let N(P) denote the set of natural numbers all of whose prime factors are in P. We say that P is Erdős-best among primitive subsets of N(P) if the inequality \sum_{n\in S} 1/(n log n) \le \sum_{n\in P} 1/(p log p) holds for every …
The prime-counting Copeland-Erdős constant
本文研究由素数计数函数值连接而成的新常数，利用 Szüsz-Volkmann 组合方法证明 Cramér 素数间隙猜想可推出该常数在任意进制下的正规性。
Let $x\geqslant 3$. For $1\leqslant n\leqslant x$ an integer, let $ω(n)$ be its number of distinct prime factors. We show that $ω(n-1)$ satisfies an Erdős-Kac type theorem whenever $ω(n)=k$ where $1\leqslant k\ll\log\log x$, thus extending a result of Halberstam.
One inequality inspired by Erdős
受Erdős证明Bertrand假设的启发，本文求解了一个正整数不等式，为证明任意正整数n≥2存在素数p满足n<p<2n提供了纯算术证明思路。
A tyro's approach to the tilted sieve: beyond the Erdős--Rankin bound
本文初等阐述倾斜筛法，证明每素数取一剩余类可覆盖长度为 x log x/((log_2 x)log_3 x) 的区间，改进 Erdős--Rankin 界，无需 Maynard 筛权重或超图覆盖定理。
In 1982, Erdös and van Lint showed an estimate for the average of the ratio of the smallest and largest prime divisor of $n$. In this note, we apply C.H. Jia's method to give an estimate for the average of positive integer power of the ratio.
We consider the distribution of the largest prime divisor of the integers in the interval $[2,x]$, and investigate in particular the mode of this distribution, the prime number(s) which show up most often in this list. In addition to giving an asymptotic formula for this mode as $x$ tends to infinity, we look at the set of those prime numbers which, for some value of $x$, occur most frequently as the largest prime divisor of the integers in the interval $[2,x]$. We find that many prime numbers never have this property. We compare the set of "popular primes," those primes which are at some poin…
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}}$.
Let $\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. Let $f(n, k) = \text{max}_{A \in C_{k}(n)}|A|$, where $|A|$ denotes the number of elements of the set $A$. Let $E_k(n)$ be the set of positive integers not exceeding $n$ which are divisible by at least one of the primes $p_{1}, \dots{}, p_{k}$, where $p_{i}$ denote the $i$th prime number. In 1962, Erdős conjectured that $f(n, k) = |E(n,k)|$ for every $n \ge p_{k}$. Recently Chen and Z…
In an attempt to resolve a folklore conjecture of Erdős regarding the non-vanishing at $s=1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\{-1, 1 \}$, Livingston conjectured the $\overline{\mathbb{Q}}$ - linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston's conjecture for composite $q \geq 4$, highlighting that a new approach is required to settles Erdős's conjecture. We also prove that the conjecture is true for prime $q \geq 3$, and indicate that more ingredients are needed to settle Erdős's co…
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.
Let $\mathcal{P}$ denote the set of all primes. In 1950, P. Erdős conjectured that if $c$ is an arbitrarily given constant, $x$ is sufficiently large and $a_1,\dots , a_t$ are positive integers with $a_1<a_2<\cdot\cdot\cdot<a_t\leqslant x$ and $t>\log x$, then there exists an integer $n$ so that the number of solutions of $n=p+a_i$ $(p\in \mathcal{P}, 1\le i\le t)$ is greater than $c$. In this note, we confirm this old conjecture of Erdős.
In this paper we obtain a new lower bound on the Erdős distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\subset \mathbb{F}_p^2$ with $|A|\le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |Δ(A)| \gg |A|^{\frac{1}{2}+\frac{149}{4214}}.$$ Our result gives a new lower bound of $|Δ{(A)}|$ in the range $|A|\le p^{1+\frac{149}{4065}}$. The main tools we employ are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, and a lower boun…
We obtain an asymptotic formula for the number of ways to represent every reduced residue class as a product of a prime and square-free integer. This may be considered as a relaxed version of a conjecture of Erdös, Odlyzko, and Sárközy.
The One-Power Erdős Conjecture for Primes $p \equiv 7 \pmod{36}$
针对素数 $p \equiv 7 \pmod{36}$，研究 Erdős 猜想：每个此类素数可写成 $2^a$ 加无平方因子数。利用分离原理将平方因子问题分解为线性问题，证明 $a \le 3$ 表示者密度为 $1-\rho_4$，并计算验证至 $10^8$。
Today, prime numbers attained exceptional situation in the area of numbers theory and cryptography. As we know, the trend for accessing to the largest prime numbers due to using Mersenne theorem, although resulted in vast development of related numbers, however it has reduced the speed of accessing to prime numbers from one to five years. This paper could attain to theorems that are more extended than Mersenne theorem with accelerating the speed of accessing to prime numbers. Since that time, the reason for frequently using Mersenne theorem was that no one could find an efficient formula for a…
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.
On the largest prime factors of shifted semiprime numbers
该研究证明了对于固定整数a和足够大的x，移位半素数乘积的最大大素因子超过x^θ，其中θ为0.5减去一个任意小的正数。
Let P(k) be the largest prime factor of the positive integer k. In this paper, we prove that the series $\sum_{n\ge 1}\frac{(\log n)^a}{P(2^n-1)}$ is convergent for each constant a<1/2, which gives a more precise form of a result of C. L. Stewart from 1977.
Let $P(X)\in\mathbb{Z}[X]$ be an irreducible, monic, quartic polynomial with cyclic or dihedral Galois group. We prove that there exists a constant $c_P>0$ such that for a positive proportion of integers $n$, $P(n)$ has a prime factor $\ge n^{1+c_P}$.
We consider the Dickman function $Ψ(x,y)$ in the limit when $\ln x/\ln y\to \infty $ and $\ln \ln x \ln y\to 0$. The asymptotic value is expressed in terms of the ratio of iterated loragithm of $x$ and $ln y$.
It was conjectured by Gauss that any negative discriminant with at least 32 genera has at least two classes of binary quadratic forms in each genus. We prove that such a discriminant cannot have a particularly large prime factor, by an argument similar to that of Baker's solution to Gauss' class number one problem.
Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on $a$, as well as the closely related problem of improving bounds $bc$, and $abc$. In particular, we prove two results. First we prove a new general bound on any prime divisor of an odd perfect number and obtain as a corollary of that bound that $$a < 2N^{\frac{1}{6}}.$$ Second, we show that $$abc < (2N)^{\frac{3}{5}}.$$ We also show how in certain circumstances these bounds …
Stewart (2013) proved that the biggest prime divisor of the $n$th term of a Lucas sequence of integers grows quicker than $n$, answering famous questions of Erdős and Schinzel. In this note we obtain a fully explicit and, in a sense, uniform version of Stewart's result.
Let $P(m)$ denote the largest prime factor of an integer $m\geq 2$, and put $P(0)=P(1)=1$. For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq 2-k}$ be the $k-$generalized Fibonacci sequence which starts with $0,...,0,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. Here, we show that if $n\geq k+2$, then $P(F_n^{(k)})>c\log\log n$, where $c>0$ is an effectively computable constant. Furthermore, we determine all the $k-$Fibonacci numbers $F_n^{(k)}$ whose largest prime factor is less than or equal to 7.
A solution to a Paul Erdos problem
本文解决了埃尔德什提出的数论问题，证明不存在素数p>5使得2!到(p-1)!模p的余数全不相同。
Prime-Power Rarefaction and a Density-One Lower Bound for Erdős Problem 400
本文通过素幂稀疏化技术，结合均匀相位分离与数字和正态阶定理，证明了 Erdős 问题 400 中函数 g_k(n) 的密度一下界与逐点上界。
For an integer $m >1$, we denote by $P(m)$ the largest prime divisor of $m$. We prove that $\limsup_{n \rightarrow +\infty} P(n!+1)/n \geqslant 1+9\log 2>7.238$, which improves a result of Stewart. More generally, for any nonzero polynomial $f(X)$ with integer coefficients, we show that $\limsup_{n \rightarrow +\infty} P(n!+f(n))/n \geqslant 1+9\log2$. This improves a result of Luca and Shparlinski. These improvements come from an additional combinatoric idea to the works mentioned above.
Let $\{p_j(n)\}_{j=1}^{ω(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. We provide details for the proof of a statement of Erdős implying that, for any function $ξ(n)$ tending to infinity with $n$, we have $$f(n):=\max_{1\leqslant j<ω(n)}\log \Big({\log p_{j+1}(n)\over \log p_j(n)}\Big)=\log_3n+O(ξ(n))$$ for almost all integers $n$.
In this paper we investigate the distribution of the number of primes which ramify in number fields of degree d <= 5. In analogy with the classical Erdos-Kac theorem, we prove for S_d-extensions that the number of such primes is normally distributed with mean and variance log log X.
Erdős-Kac theorems for discriminants of number fields
本文证明了当 G 为阿贝尔群时，随机 G-扩张中分歧素数个数的中心极限定理，推广了 Lemke Oliver 和 Thorne 在 G=S_d (2≤d≤5) 情形的工作，并首次给出不同素数处分歧事件不独立的例子。
A Ceiling Continued Fraction Approach to the Erdős-Straus Conjecture: Heuristic finiteness of counterexamples
提出取整连分数（FCT）框架构造Erdős-Straus猜想的三项埃及分数表示，利用移位整数p+i的除数结构而非同余技巧，推导出失败概率的超多项式上界，结合Borel-Cantelli引理启发式证明反例有限，并在10^17、10^52和10^131附近的10^9个素数上验证无反例。
On an iterated arithmetic function problem of Erdos and Graham
本文研究 Erdős 和 Graham 提出的迭代算术函数方程，将 r=2 情形归结为求解 ϕ(n)+ϕ(n+ϕ(n))=n，并给出显式解集，同时证明其余可能解依赖于极罕见的大素数条件。
We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bretèche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman's sieve method. To prove the Type II estimate we use the bounds of Deshouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$
Let $M(x)$ denote the median largest prime factor of the integers in the interval $[1,x]$. We prove that $$M(x)=x^{\frac{1}{\sqrt{e}}\exp(-\text{li}_{f}(x)/x)}+O_ε(x^{\frac{1}{\sqrt{e}}}e^{-c(\log x)^{3/5-ε}})$$ where $\text{li}_{f}(x)=\int_{2}^{x}\frac{\{x/t\}}{\log t}dt$. From this, we obtain the asymptotic $$M(x)=e^{\frac{γ-1}{\sqrt{e}}}x^{\frac{1}{\sqrt{e}}}(1+O(\frac{1}{\log x})),$$ where $γ$ is the Euler Mascheroni constant. This answers a question posed by Martin, and improves a result of Selfridge and Wunderlich.
The largest prime factor of an irreducible cubic polynomial
将Heath-Brown关于$n^3+2$存在大于$n^{1+c}$的素因子的结论推广到任意首一不可约三次整系数多项式，并给出依赖于多项式的指数$c_p$。
An improvement on the largest prime factor of $n^2+1$
本研究改进了$n^2+1$最大素因子的增长下界，将此前的$(\nlog_2 n)^2/\nlog_3 n$提升至$(\nlog_2 n)^2/\nlog_4 n$，并证明该下界接近紧时同样适用于$n^2+1$的多个素因子。
We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations. Using this recursion we can estimate the numbers of a gap or of a constellation that occur between a prime and its square. This recursion also has explicit implications for open questions about gaps between prime numbers, including three questions posed by Erdös and Turán.
On the largest prime factor of quadratic polynomials
本文基于 Grimmelt 和 Merikoski 的近期结果，证明了 $n^2+1$ 的最大素因子无穷多次大于 $x^{1.317}$，并给出了满足 $n \leqslant x$ 且 $n^2+1$ 具有本原因子的整数 $n$ 个数的新上界。
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