Artyom Radomskii
We obtain a lower bound for \[ \#\{x/2< p_{n}\leq x:\ p_n \equiv\ldots\equiv p_{n+m}\equiv a\text{ (mod $q$)},\ p_{n+m} - p_{n}\leq y\}, \] where $p_{n}$ is the $n^{\text{th}}$ prime.
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We show that a positive proportion of all gaps between consecutive primes are small gaps. We provide several quantitative results, some unconditional and some conditional, in this flavour.
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
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.
We prove that a positive proportion of the gaps between consecutive primes are short gaps of length less than any fixed fraction of the average spacing between primes.
We show that for every positive integer $k$, there exist $k$ consecutive primes having the property that if any digit of any one of the primes, including any of the infinitely many leading zero digits, is changed, then that prime becomes composite.
Making use of covering systems and a theorem of D. Shiu, the first and second authors showed that for every positive integer $k$, there exist $k$ consecutive widely digitally delicate primes. They also noted that for every positive integer $k$, there exist $k$ consecutive primes which are Brier numbers. We show that for every positive integer $k$, there exist $k$ consecutive primes that are both widely digitally delicate and Brier numbers.
Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+ε}p_n)$ where $ε> 0$, in the context of the results we have obtained in our paper.
We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.
Using as the working hypothesis of an evaluation of the difference between primes $p_{n+1} - p_n = O(\sqrt{p_n})$ we represent in detail the proofs of Legendre's and Oppermann's conjectures.
We investigate the consecutive primes $p$ and $q$ ($p > q$) for which there exists a pair of natural numbers $(x,y)$ such that $p^x-q^y$ is a perfect square and make some conjectures.
We give an explicit form of Ingham's Theorem on primes in the short intervals, and show that there is at least one prime between every two consecutive cubes $x\sp{3}$ and $(x+1)\sp{3}$ if $\log\log x\ge 15$.
We derive heuristically formula for the $k$--moments $M_k(x)$ of the gaps between consecutive primes$<x $ represented directly by $x$$π(x)$ --- the number of primes up to: $M_k(x)= Γ(k+1)x^k/π^{k-1}(x)+\mathcal{O}(x)$, We illustrate obtained results by computer data.
We show by an inclusion-exclusion argument that the prime $k$-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect of a theorem of Gallagher that the prime $k$-tuple conjecture implies that the prime numbers are distributed in a Poisson distribution around their average spacing.
We show that the product of two consecutive Fibonacci (respectively Lucas) numbers is divisible by the sum of their indices if this sum is a prime number different from 5 and in the form (4r+1)(respectively (4r+3)).
For any positive integer $k$, we show that infinitely often, perfect $k$-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$ c_k \frac{\log p \log_2 p \log_4 p}{(\log_3 p)^2}, $$ where $p$ is the smaller of the two primes.
The goal of the present paper is to present a method of proving of Diophantine inequalities with primes through the use of auxiliary inequalities and available evaluations of the difference between consecutive primes. We study the Legendre - Ingham's problem on primes in intervals $((n - 1)^k, n^k)$ and also a problem on primes in intervals $(\frac{k-1}{k}n, \frac{k}{k-1}n)$ when $k$ is a real number. A number of the new results including an alternative proof of Ingham's theorem with the effectively computable constant and also Ingham's theorem with two primes are proved.
The behaviour of a certain additive function in large intervals between consecutive primes
本文研究连续素数之间大区间内特定整数素因子所决定的加性函数的行为，结合素数大间隙相关结果、Erdős–Kac 定理及 Kubilius 模型得到主要结论。
Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a fixed positive constant, for which the interval $(p_n, p_{n+1})$ contains the $k$-th power of a prime number.
This paper updates the explicit interval estimate for primes between consecutive powers. It is shown that there is least one prime between $n^{155}$ and $(n+1)^{155}$ for all $n\geq 1$. This result is in part obtained with a new explicit version of Goldston's 1983 estimate for the error in the truncated Riemann--von Mangoldt explicit formula.
Bounded gaps and perfect power gaps in sequences of consecutive primes
本文研究连续素数间隙中是否存在较大间隙或完美幂间隙，并探讨相关数学问题。
We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $π(x)$. As the application of this formula we formulate 7 conjectures, among others for the maximal gap between two consecutive primes smaller than $x$, for the generalized Brun's constants and the first occurrence of a given gap $d$. Also the leading term $\log \log(x)$ in the prime harmonic sum is reproduced from our guesses correctly. These conjectures are supported by the computer data.
In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach we have been able to prove some theorems, including Ingham's exponential theorem as well as some new results. Diophantine inequalities and their connection with Cramer's and Andrica's conjectures are also discussed.
Minimal zero-free regions for results on primes between consecutive perfect $k$th powers
该研究计算了确保连续完美k次幂间存在素数的最小无零区域，并证明了在连续86次幂间总存在素数，同时识别了一个特定整数序列在连续70次幂间也总存在素数。
We derive heuristically approximate formulas for the negative $k$--moments $M_{-k}(x)$ of the gaps between consecutive primes$<x $ represented directly by $π(x)$ --- the number of primes up to $x$. In particular we propose an analytical formula for the sum of reciprocals of gaps between consecutive primes $<x : ~ M_{-1}(x)\sim \frac{π^2(x)}{x-2π(x)}\log\Big(\frac{x}{2π(x)}\Big) \sim x \log \log(x)/\log^2(x)$. We illustrate obtained results by the enormous computer data up to $x=4\times 10^{18}$.
While the sequence of primes is very well distributed in the reduced residue classes (mod $q$), the distribution of pairs of consecutive primes among the permissible $ϕ(q)^2$ pairs of reduced residue classes (mod $q$) is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared to numerical data, and the observed fit is very good.
We show that \[\sum_{\substack{p_n\le x\\ p_{n+1}-p_n\ge\sqrt{p_n}}}(p_{n+1}-p_n)\ll_{\varepsilon} x^{3/5+\varepsilon}\] for any fixed $\varepsilon>0$. This improves a result of Matomäki, in which the exponent was $2/3$.
Let $α, β\geq 0$ and $α+ β< 1$. In this short note, we show that $\liminf_{n \to \infty} p_n^β(p_{n+1}^α- p_n^α) = 0$, where $p_n$ is the $n$th prime. This notes an improvement over results of Sándor and gives additional evidence towards a conjecture of Andrica. This follows directly from recent results on prime pairs from Maynard, Tao, Zhang.
The following is proven using arguments that do not revolve around the Riemann Hypothesis or Sieve Theory. If $p_n$ is the $n^{\rm th}$ prime and $g_n=p_{n+1}-p_n$, then $g_n=O({p_n}^{2/3})$.
In this note, we study the problem of existence of sequences of consecutive 1's in the periodic part of the continued fractions expansions of square roots of primes. We prove unconditionally that, for a given $N\gg 1$, there are at least $N\log^{-3/2}N$ prime numbers $p\leq N$ such that the continued fraction expansion of $\sqrt{p}$ contains three consecutive 1's on the beginning of the periodic part. We also present results of our computations related to the considered problem and some related problems, formulate several open questions and conjectures and get some results under the assumption…
Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.
For each of the functions $f \in \{ϕ, σ, ω, τ\}$ and every natural number $k$, we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots < f(p_{n+k}-1)$, and similarly for $f(p_n-1) > f(p_{n+1}-1) > \dots > f(p_{n+k}-1)$. We also answer some questions of Sierpiński on the digit sums of consecutive primes. The arguments make essential use of Maynard and Tao's method for producing many primes in intervals of bounded length.
Update: This work reproduces an earlier result of Peck, which the author was initially unaware of. The method of the proof is essentially the same as the original work of Peck. There are no new results. We show that the sum of squares of differences between consecutive primes $\sum_{p_n\le x}(p_{n+1}-p_n)^2$ is bounded by $x^{5/4+ε}$ for $x$ sufficiently large and any fixed $ε>0$. The proof relies on utilising various mean-value estimates for Dirichlet polynomials.
Let $p_n$ denote the $n$-th prime number, and let $d_n=p_{n+1}-p_{n}$. Under the Hardy--Littlewood prime-pair conjecture, we prove \begin{align*} \sum_{n\le X}\frac{\log^αd_n}{d_n} \sim\begin{cases} \frac{X\log\log\log X}{\log X}~\qquad\quad~ &α=-1,\\ \frac{X}{\log X}\frac{(\log\log X)^{1+α}}{1+α}\qquad &α>-1, \end{cases} \end{align*} and establish asymptotic properties for some series of $d_n$ without the Hardy--Littlewood prime-pair conjecture.
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…
On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares
本文通过改进Dudek和Johnston的结果，证明了每对连续平方之间存在一个具有最多3个素因子的整数，从而在素数分布理论中取得进展。
Recently Oliver and Soundararajan made conjectures based on computational enumerations about the frequency of occurrence of pairs of last digits for consecutive primes. By studying Eratosthenes sieve, we have identified discrete dynamic systems that exactly model the populations of gaps across stages of Eratosthenes sieve. Our models provide some insight into the observed biases in the occurrences of last digits in consecutive primes, and the models suggest that the biases will ultimately be reversed for large enough primes. The exact model for populations of gaps across stages of Eratosthenes…
We show that $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{1/2}}} (p_{n+1} - p_n) \ll x^{0.57+ε}$$ and $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{0.45}}} (p_{n+1} - p_n) \ll x^{0.63+ε},$$ where $p_n$ is the $n$th prime number. The proof combines Heath-Brown's recent work with Harman's sieve, improving and extending his results. We give applications of the results to prime-representing functions, binary digits of primes and approximation of reals by multiplicative functions.
Fix irrational numbers $α,\hatα>1$ of finite type and real numbers $β,\hatβ\ge 0$, and let $B$ and $\hat B$ be the Beatty sequences $$ B:=(\lfloorαm+β\rfloor)_{m\ge 1}\quad\text{and}\quad\hat B:=(\lfloor\hatαm+\hatβ\rfloor)_{m\ge 1}. $$ In this note, we study the distribution of pairs $(p,p^\sharp)$ of consecutive primes for which $p\in B$ and $p^\sharp\in\hat B$. Under a strong (but widely accepted) form of the Hardy-Littlewood conjectures, we show that $$ \big|\{p\le x:p\in B\text{ and }p^\sharp\in\hat B\}\big|=(α\hatα)^{-1}π(x)+O\big(x(\log x)^{-3/2+ε}\big), $$ where $π(x)$ is the prime cou…
Let $p_n$ denote the $n$-th prime number, $\{q_n\}$ be a sequence of positive numbers and $x\in\mathbb{R}$. In this note we prove that the inequality $$q_n p_{n+1}^{x}-q_{n+1}p_{n}^{x}<p_{n}^{x}p_{n+1}^{x-1}, $$ holds for infinitely many values of $n$. As it is shown, the key ingredient to obtain this behaviour is a consequence of an extension of the Kummer's characterization of convergent series of positive terms.
The most common difference that occurs among the consecutive primes less than or equal to $x$ is called a jumping champion. Occasionally there are ties. Therefore there can be more than one jumping champion for a given $x$. In 1999 A. Odlyzko, M. Rubinstein, and M. Wolf provided heuristic and empirical evidence in support of the conjecture that the numbers greater than 1 that are jumping champions are 4 and the primorials 2, 6, 30, 210, 2310,... As a step towards proving this conjecture they introduced a second weaker conjecture that any fixed prime $p$ divides all sufficiently large jumping c…
In recent work, we considered the frequencies of patterns of consecutive primes $\pmod{q}$ and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to bot…
Nonexistence of Consecutive Powerful Triplets Around Cubes with Prime-Square Factors
本文证明了围绕具有素数平方因子的立方体的连续强力三元组不存在，结合模算术、p-adic赋值、Thue方程和椭圆曲线理论，推进了Erdős-Mollin-Walsh猜想的研究。
Consecutive primes in tuples
本文研究了元组中连续素数的分布，证明了存在无限多个n使得H(n)包含至少m个连续素数，并构造了具有递增或递减间隙的素数字符串。
Consecutive integers free of certain prime factors
本文证明了Erdős猜想：存在无穷多个k，使得最小整数n>2k满足(n-k)...(n-1)不被(k,2k)内任何素数整除，且n_k > e^{log^2 k/(20 log log k)}。
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…
Exact Frequencies of Consecutive Quadratic Residue and Nonresidue Patterns Modulo a Prime
研究模奇素数\(p\)时长度为二和三的连续二次剩余与非剩余模式的精确频率，利用有限域特征和基本性质推导计数公式，对的频率由\(p \pmod 4\)分类，三元组频率依赖\(p \pmod 8\)及雅可比和\(T_p\)。
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.
A note on zero-density approaches for the difference between consecutive primes
本研究推广了短区间素数相关的Ingham定理与Heath-Brown和Iwaniec的结果，明晰了无零点区域、零密度估计和短区间素数的关联，还在密度假设下改进了素数定理成立的短区间范围。
Let $m$ be any positive integer and let $δ_1,δ_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=δ_1\ \quad\text{and}\ \quad\left(\frac{p_{n+j}}{p_{n+i}}\right)=δ_2$$ for all $0\le i<j\le m$, where $p_k$ denotes the $k$-th prime, and $(\frac {\cdot}p)$ denotes the Legendre symbol for any odd prime $p$. We also prove that under the Generalized Riemann Hypothesis there are infinitely many positive integers $n$ such that $p_{n+i}$ is a primitive root modulo $p_{n+j}$ for any di…
For any real $x$ and any integer $k\ge1$, we say that a set $\mathcal{D}_{k}$ of $k$ distinct integers is a $k$-tuple jumping champion if it is the most common differences that occurs among $k+1$ consecutive primes less than or equal to $x$. For $k=1$, it's known as the jumping champion introduced by J. H. Conway. In 1999 A. Odlyzko, M. Rubinstein, and M. Wolf announced the Jumping Champion Conjecture that the jumping champions greater than 1 are 4 and the primorials 2, 6, 30, 210, 2310,.... They also made a weaker and possibly more accessible conjecture that any fixed prime $p$ divides all su…
Residue Class Patterns of Consecutive Primes
研究连续素数的剩余类模式，通过结合狄利克雷定理与Shiu定理，修改梅纳德 - 陶筛法并结合埃尔德什 - 兰金类型构造进行证明，得出在特定条件下关于长度为\(m\)的剩余类模式无限次出现的相关结论及数量估计。
Consecutive Piatetski-Shapiro primes based on the Hardy-Littlewood conjecture
本文基于Hardy-Littlewood猜想，研究连续素数对分别属于两个Piatetski-Shapiro序列的分布，提出计数函数猜想并给出启发式论证，还证明了奇异级数加权平均的命题。
Arbitrarily long strings of consecutive primes in special sets
本文研究一类由广义多项式小数部分落入非空开集所定义的自然数集合，证明其中存在任意长的连续素数串，揭示了此类特殊集合在素数分布上的强正则性。
I present a new property of prime numbers that leads to a generalization of Cramer's conjecture. The study of the gap between consecutive primes is treated as a special case of the gap between consecutive terms of sequences having a certain property called pseudo equidistribution. Both rigorous as well as heuristic arguments are given in support of the generalized Cramer's conjectures.
Rough numbers between consecutive primes
本文用筛法证明相邻素数间隙中几乎总存在最小素因子不小于间隙长度的数，并给出例外间隙数的上界及在猜想下的渐近式。
Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen consecutively from a sequence in a finite field of odd prime characteristic. We study the arithmetic of such sequences, including bounds for the largest degree of irreducible factors, the number of irreducible factors, as well as for the number of such sequences of fixed length in which all the polynomials are irreducible.
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.
For $q$ an odd prime power, we prove that there are always four consecutive primitive elements in the finite field $\mathbb{F}_{q}$ when $q> 2401$.
An inequality involving primes and the product of primes
本文证明了一个涉及素数及连续素数乘积的不等式。
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.
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