On sequences of integers with small prime factors
C. L. Stewart
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。
我们证明，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。
We show that the difference between consecutive terms in sequences of integers whose greatest prime factor grows slowly tends to infinity.
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Exploring the Representation of Large Positive Integers as Sums of Prime Powers and Integer Powers: Analysis with Positive Density Subsets
本文利用转换原理研究大正整数表示为素数幂与整数幂之和的问题，其中素数和整数幂分别取自正密度子集，拓展了加法数论中的表示理论。
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.
In this paper, we develop Furstenberg's proof of infinity of primes, and prove several results about prime divisors of sequences of integers, including the celebrated Schur's theorem. In particular, we give a simple proof of a classical result which says that a non-degenerate linear recurrence sequence of integers of order k>1 has infinitely many prime divisors.
For a fixed integer $r\ge1$, we say $k$-tuple integers $(x_1,\ldots,x_k)$ are relatively $r$-prime if there exists no prime $p$ such that all $k$ integers is multiple of $p^r$. Benkoski proved that the number of relatively $r$-prime $k$-tuple integers in $[1,x]^k$ is $x^k/ζ(rk)+$(Error term). We showed that the exact order of error term is $x^{k-1}$ for $rk\ge3$ and $k\not=1$.
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.
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 $b$.
We study the problem of representing integers as sums of prime numbers from a fixed Beatty sequence $B_{α,β}$, where $α>1$ is irrational and of finite type.
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 $X^4+aX+b\in\z[X]$.
Uniformly for small $q$ and $(a,q)=1$, we obtain an estimate for the weighted number of ways a sufficiently large integer can be represented as the sum of a prime congruent to $a$ modulo $q$ and a square-free integer. Our method is based on the notion of local model developed by Ramaré and may be viewed as an abstract circle method.
The set of integers which can be written as the sum of four prime cubes has lower density at least $0.009664$. This improves earlier bounds of $0.003125$ by Ren and $0.005776$ by Liu.
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.
The average number of representations of an integer as a sum of two prime powers over multiples of a fixed integer
本文扩展Ikeda和Suriajaya(2025)的结果，研究固定整数q的倍数上整数n作为两个素数k次幂之和的表示数的渐进行为。
We prove that every positive integer $n$ which is not equal to $1$, $2$, $3$, $6$, $11$, $30$, $155$, or $247$ can be represented as a sum of a squarefree number and a prime not exceeding $\sqrt{n}$.
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.
Let $N(x,y)$ denote the number of integers $n\le x$ which are divisible by a shifted prime $p-1$ with $p>y$, $p$ prime. Improving upon recent bounds of McNew, Pollack and Pomerance, we establish the exact order of growth of $N(x,y)$ for all $x\ge 2y\ge 4$.
We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for $\de$ arbitrarily small and positive, the nearest neighbor spacings between integers $n$ with $|\om(n)-\log_2 n|\le (\log_2 n)^{\de}$ obey the Poisson distribution law.
Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best possible.
On the representation of integer as sum of a square-free number and a prime of special type
本文证明存在无穷多个整数可表示为一个无平方因子整数与一个满足特定丢番图不等式条件的素数之和，其中系数α为无理数。
The average number of representations of an integer as a sum of three or more primes, over multiples of a fixed integer
该论文将Ikeda和Suriajaya的结果推广至任意$j\ge 3$个素数之和的表示数平均值，证明一般情形下渐近公式成立，并引入广义哥德巴赫问题奇异级数的平均值作为新工具。
Let $α> 0$ be any fixed non-integer, $I$ be any subinterval of $[0; 1)$. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for the set of primes $p$ satisfying the condition $\{ p^α \} \in I$. This strengthens the previous result of Gritsenko and Zinchenko.
Prime Gaps In The Gaussian Integers
本文定义了高斯整数中的素数间隙，并利用数值方法证明了其渐近上界为 O(log^2|p_n|)。
We show that for integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is $C \log \log n$, where $C=(1-e^{-γ})^{-1} = 2.280...$ and $γ$ is Euler's constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers.
A Congruence for Sums of Integer Powers Modulo Products of Distinct Primes
本文证明了在特定条件下，不同素数乘积模下的整数幂和满足一种同余关系。
On the representation of large even integers as the sum of eight primes from positive density sets
本文证明若素数子集 A 在素数中的下密度大于 1/2，则每个充分大的偶数均可表示为 A 中八个素数之和，且常数 1/2 是最佳的。
We investigate the distribution of the digits of quotients of randomly chosen positive integers taken from the interval $[1,T]$, improving the previously known error term for the counting function as $T\to+\infty$. We also resolve some natural variants of the problem concerning points with prime coordinates and points that are visible from the origin.
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 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。
Let $r\ge k\ge 2$ be fixed positive integers. Let $\varrho_{r,k}$ denote the characteristic function of the set of $r$-tuples of positive integers with $k$-wise relatively prime components, that is any $k$ of them are relatively prime. We use the convolution method to establish an asymptotic formula for the sum $\sum_{n_1,\ldots,n_r\le x} \varrho_{r,k}(n_1,\ldots,n_r)$ by elementary arguments. Our result improves the error term obtained by J. Hu (2013).
Arithmetic progressions of integers that are relatively prime to their digital sums
本文研究了$b$-反尼文数等差数列的最大长度，探讨了其在不同基底下的性质和特征。
On Maximal Values of Gronwall Numbers for Integers with Given Greatest Prime Factor and Remainder in Modified Mertens Formula
研究整数最大格罗旺数与修改后的梅特恩公式中素数倒数和余数之间的关系，不依赖黎曼假设，提出新的极限关系。
Representing integers as sums of mixed powers of primes
该研究证明了每个足够大的奇整数可以表示为素数的混合次幂之和，其中包含平方、立方和更高次幂的组合。
Given an integer n>1, it is a classical Diophantine problem that whether n can be written as a sum of two rational cubes. The study of this problem, considering several special cases of n, has a copious history that can be traced back to the works of Sylvester, Satgé, Selmer etc. and up to the recent works of Alpöge-Bhargava-Shnidman. In this article, we consider the cube sum problem for cube-free integers n which has two distinct prime factors none of which is 3.
We continue to study the distribution of prime numbers $p$, satisfying the condition $\{ p^α \} \in I \subset [0; 1)$, in arithmetic progressions. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for $0 < α< 1/9$ with the level of distribution $θ= 2/5 - (3/5) α$, which improves the previous result corresponding to $θ\leq 1/3$.
We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$ [5] respectively. We use the same technique to study the corresponding problem for sums of just $k$ perfect $k$-th powers of primes.
In this paper we show that for every positive integer $n$ there exists a prime number in the interval $[n,9(n+3)/8]$. Based on this result, we prove that if $a$ is an integer greater than 1, then for every integer $n>14.4a$ there are at least four prime numbers $p$, $q$, $r$, and $s$ such that $n<ap<3n/2<aq<2n$ and $n<r<3n/2<s<2n$. Moreover, we also prove that if $m$ is a positive integer, then for every positive integer $n>14.4/(|\sqrt[m]{1.5}|-1)^m$ there exist a positive integer $a$ and a prime number $s$ such that $n<a^m<3n/2<s<2n$, as well as the fact that for every positive integer $n>14…
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。
It is shown that every sufficiently large even integer is a sum of two primes and exactly 13 powers of 2. Under the Generalized Rieman Hypothesis one can replace 13 by 7. Unlike previous work on this problem, the proof avoids numerical calculations with explicit zero-free regions of Dirichlet L-functions. The argument uses a new technique to bound the measure of the set on which the exponential sum formed from powers of 2 is large.
We study whether sufficiently large integers can be written in the form cp+T_x, where p is either zero or a prime congruent to r mod d, and T_x=x(x+1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2^ap-r)/m+T_x with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with related analysis and numerical data. One of our conjectures states that each natural number not equal to 216 can be written in the form p+T_x with x an integer and p a prime or zero; another conjecture asser…
Asymptotic counting of integers with prime factors $p_{r^a s^b}$
本文给出具有形如 $p_{r^a s^b}$ 的素数因子的整数的渐近计数公式，误差为 $o(1)$，并解决了 $(2,3)$ 情形下序数编码的计数问题。
We prove an asymptotic formula for the number of integers $\leq x$ which can be written as the product of $k ~(\geq 2)$ distinct primes $p_1\cdots p_k$ with each prime factor in an arithmetic progression $p_j\equiv a_j \bmod q$, $(a_j, q)=1$ $(q \geq 3, 1\leq j\leq k)$. For any $A>0$, our result is uniform for $2\leq k\leq A\log\log x$. Moreover, we show that, there are large biases toward certain arithmetic progressions $(a_1 \bmod q, \cdots, a_k \bmod q)$, and such biases have connections with Mertens' theorem and the least prime in arithmetic progressions.
We consider the problem of finding small prime gaps in various sets of integers $\mathcal{C}$. Following the work of Goldston-Pintz-Yildirim, we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Letting $q_n$ denote the $n$-th prime in $\mathcal{C}$, we will establish that for any small constant $ε>0$, the set $\left\{q_n| q_{n+1}-q_n \leq ε\log n \right\}$ constitutes a positive proportion of all prime numbers. Using the techniques developed by Maynard and Tao we will also demonstrate that $\mathcal{C}$ has bounded prime gaps. Specific examples,…
In this paper, we consider the simultaneous representation of pairs of sufficiently large integers. We prove that every pair of large positive odd integers can be represented in the form of a pair of one prime, four cubes of primes and 231 powers of 2.
Goldston, Pintz and Yıldırım have shown that if the primes have `level of distribution' $θ$ for some $θ>1/2$ then there exists a constant $C(θ)$, such that there are infinitely many integers $n$ for which the interval $[n,n+C(θ)]$ contains two primes. We show under the same assumption that for any integer $k\ge 1$ there exists constants $D(θ,k)$ and $r(θ,k)$, such that there are infinitely many integers $n$ for which the interval $[n,n+D(θ,k)]$ contains two primes and $k$ almost-primes, with all of the almost-primes having at most $r(θ,k)$ prime factors. If $θ$ can be taken as large as $1-ε$, …
Let c > 0.55. Every large n can be written in the form p +ab, where p is prime, a and b are significantly smaller than x^1/2 and ab is less than n^c. This strengthens a result of Heath-Brown, which has the requirement c>3/4. We introduce the idea of 'intersecting two sieves' as a tool in the proof of this result.
Let $f(n)$ be the number of distinct exponents in the prime factorization of the natural number $n$. We prove some results about the distribution of $f(n)$. In particular, for any positive integer $k$, we obtain that $$ \#\{n \leq x : f(n) = k\} \sim A_k x $$ and $$ \#\{n \leq x : f(n) = ω(n) - k\} \sim \frac{B x (\log \log x)^k}{k! \log x} , $$ as $x \to +\infty$, where $ω(n)$ is the number of prime factors of $n$ and $A_k, B > 0$ are some explicit constants. The latter asymptotic extends a result of Aktaş and Ram Murty about numbers having mutually distinct exponents in their prime factoriza…
Let $p$ be a prime. We define $S(p)$ the smallest number $k$ such that every positive integer is a sum of at most $k$ squares of integers that are not divisible by $p$. In this article, we prove that $S(2)=10$, $S(3)=6$, $S(5)=5$, and $S(p)=4$ for any prime $p$ greater than $5$. In particular, it is proved that every positive integer is a sum of at most four squares not divisible by $5$, except the unique positive integer $79$.
Toda primes
该研究提出Toda素数的定义，猜想每个正整数至少有两个Toda素数，给出至少存在一个的部分证明，并探讨其与Bernoulli数分母的关联及对Sophie Germain素数的推广。
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.
Exponential sums over integers without large prime divisors
本文通过更精确地利用无大素因子整数的因数分解并结合Type I和Type II界，改进了此类整数及其 $ν$ 次幂上的指数和的现有界限。
For a real parameter $r$, the RSA integers are integers which can be written as the product of two primes $pq$ with $p<q\leq rp$, which are named after the importance of products of two primes in the RSA-cryptography. Several authors obtained the asymptotic formulas of the number of the RSA integers. However, the previous results on the number of the RSA integers were valid only in a rather restricted range of the parameter $r$. Dummit, Granville, and Kisilevsky found some bias in the distribution of products of two primes with congruence conditions. Moree and the first author studied some sim…
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 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.
Under Cramér's conjecture concerning the prime numbers, we prove that for any $x>1$, there exists a real $A=A(x)>1$ for which the formula $[A^{n^x}]$ (where $[]$ denotes the integer part) gives a prime number for any positive integer $n$. Under the same conjecture, we also prove that for any $ε>0$, there exists a positive real number $B$ for which the formula $[B.{n!}^{2+ε}]$ gives a prime number for any sufficiently large positive integer $n$.
In 1951, Linnik proved the existence of a constant $K$ such that every sufficiently large even number is the sum of two primes and at most $K$ powers of 2. Since then, this style of approximation has been considered for problems similar to the Goldbach conjecture. One such problem is the representation of a sufficiently large even number as a sum of four squares of primes and at most $k$ powers of two. In 2014, Zhao proved this to be true with $k = 46$. In this paper, we reduce this to $k = 31$.
On certain sums involving the largest prime factor over integer sequences
本文研究了整数序列中最大素因子相关的求和问题，推导了两个求和的渐进行为。
Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\ P(n)>\sqrt{x}}}f(n)\bigg|\leq\sqrt{x}(\log\log x)^{1/4+\varepsilon},$$ where $P(n)$ stands for the largest prime factor of $n$. This gives an indication of the almost sure size of the largest fluctuations of $f$.
Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions.
We define an Artin prime for an integer $g$ to be a prime such that $g$ is a primitive root modulo that prime. Let $g\in \mathbb{Z}\setminus\{-1\}$ and not be a perfect square. A conjecture of Artin states that the set of Artin primes for $g$ has a positive density. In this paper we study a generalization of this conjecture for the primes produced by a polynomial and explore its connection with the problem of finding a fixed integer $g$ and a prime producing polynomial $f(x)$ with the property that a long string of consecutive primes produced by $f(x)$ are Artin primes for $g$. By employing so…
The well-known Lagrange's four-square theorem states that any integer $n\in\mathbb{N}=\{0,1,2,...\}$ can be written as the sum of four squares. Recently, Z.-W. Sun investigated the representations of $n$ as $x^2+y^2+z^2+w^2$ with certain linear restrictions involving the integer variables $x,y,z,w$. In this paper, via the theory of quadratic forms, we further study the representations $n=x^2+y^2+z^2+w^2$ (resp., $n=x^2+y^2+z^2+2w^2$) with certain linear restrictions involving primes. For example, we obtain the following results: (i) Each positive integer $n>1$ can be written as $x^2+y^2+z^2+2w…
In RSA cryptography numbers of the form $pq$, with $p$ and $q$ two distinct proportional primes play an important role. For a fixed real number $r>1$ we formalize this by saying that an integer $pq$ is an RSA-integer if $p$ and $q$ are primes satisfying $p<q\le rp$. Recently Dummit, Granville and Kisilevsky showed that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p, q$ both prime, satisfy $p\equiv q\equiv 3\pmod{4}$. In this paper we investigate this phenomenon for RSA-integers. We establish an analogue of a strong form of the prime number theorem wit…
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