Christian Axler
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
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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。
Based on new explicit estimates for the prime counting function, we improve the currently known estimates for the particular sequence $C_n = np_n - \sum_{k \leq n}p_k$, $n \geq 1$, involving the prime numbers.
In this paper we consider the equation $[p^{c}] + [m^{c}] = N$, where $N$ is a sufficiently large integer, and prove that if $1 < c < \frac{29}{28}$, then it has a solution in a prime $p$ and an almost prime $m$ with at most $\left[ \frac{52}{29 - 28 c}\right] + 1 $ prime factors.
In this note we give some identities which involve the Mertens function M(n). Our proofs are combinatorial with relatively prime subsets as a main tool.
Goldbach-Linnik type problems involving one prime, four prime cubes and powers of 2
研究Goldbach-Linnik型问题，证明任意一对充分大奇数均可由一个素数、四个素数立方及48个2的幂表示。
Let $η$ be a quadratic irrationality. The variant of a ternary problem of Goldbach involving primes such that $a<\{ηp\}<b$, where $a$ and $b$ are arbitrary numbers of the interval $(0,1)$, solved in this paper.
Let $η$ be a quadratic irrationality. The variant of Hua Loo Keng's problem involving primes such that $a<\{ηp^2\}<b$, where $a$ and $b$ are arbitrary real numbers of the interval $(0,1)$, solved in this paper.
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 $n$.
We prove some properties of the sequence $\{a_n\}_{n\ge1}$ defined by $a_n=π(n)-π\bigl(\textstyle\sum_{k=1}^{n-1}a_k\bigr).$
A sharp asymptotic formula for the sum of reciprocals of $π(n)$ is derived, where $π(x)$ is the number of primes not exceeding $x$. This result improves the previous results of De Koninck--Ivić and L. Panaitopol.
Assume that $N$ is a sufficiently large positive number. In this paper we show that for a small constant $\varepsilon>0$, the logarithmic inequality \begin{equation*} \big|p_1\log p_1+p_2\log p_2+p_3\log p_3-N\big|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3$.
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 $\frac{51}{19}ω(N)-\frac{46}{19} \leq Ω(N)$.
We consider the equation $[p_{1}^{c}] + [p_{2}^{c}] + [p_{3}^{c}] = N$, where $N$ is a sufficiently large integer, and prove that if $1 < c < \frac{17}{16}$, then it has a solution in prime numbers $p_{1}$, $p_{2}$, $p_{3}$ such that each of the numbers $p_{1} + 2$, $p_{2} + 2$, $p_{3} + 2$ has at most $\left [ \frac{95}{17 - 16c} \right ]$ prime factors, counted with the multiplicity.
We prove that there are infinitely many solutions of $$ |λ_0+λ_1p+λ_2P_r|<p^{-τ}, $$ where $r=3,$ $τ=\frac1{118}$, and $λ_0$ is an arbitrary real number and $λ_1,λ_2\in\BR$ with $λ_2\neq0$ and $0>\frac{λ_1}{λ_2}$ not in $\mathbb{Q}$. This improves a result by Harman. Moreover, we show that one can require the prime $p$ to be of the form $\floor{n^c}$ for some positive integer $n$, i.e. $p$ is a Piatetski-Shapiro prime, with $r=13$ and $τ=ρ(c),$ a constant explicitly determined by $c$ supported in $\left(1, 1+\frac1{149}\right].$
On an additive problem involving fractional powers with one prime and an almost prime variables
本文研究分数次幂整数部分的二元加法表示，证明在指数范围内充分大整数可表为素数与素因子个数有界的殆素数之和，并改进已有结果。
General Solution of the Diophantine equation involving Mersenne Prime
该研究针对含梅森素数的指数丢番图方程$M_p^{x} + (M_q + 1)^{y}= (lz)^2$展开求解，给出了方程的解的相关示例与无解的情形，丰富了梅森素数相关丢番图方程的研究成果。
We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists $k\in\{0,\ldots,n\}$ such that $n+k$ and $n+k^2$ are both prime. (ii) Each integer $n>1$ can be written as $x+y$ with $x,y\in\{1,2,3,\ldots\}$ such that $x+ny$ and $x^2+ny^2$ are both prime. (iii) For any rational number $r>0$, there are distinct primes $q_1,\ldots,q_k$ with $r=\sum_{j=1}^k1/(q_j-1)$. (iv) Every $n=4,5,\ldots$ can be written as $p+q$, where $p$ is …
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…
We give estimates from below for the greatest prime factor of the n-th term of a binary recurrence sequence.
The Euler's totient function $ φ(n) $ counts the positive integers up to a given integer $ n$ that are relatively prime to $ n $. We solve a problem due to Lehmer that there is no composite number $ n $ such that $ φ(n)\mid n-1 $. We also note that there are infinitely many primes that are not Sophie Germain primes.
We show that if besides the primes some other sequences (involving the Liouville function and the primes) have a common distribution level exceeding 0.7231 then for any positive even integer $h$ there are arbitrarily long arithmetic progressions of primes $p$ such that $p+h$ is also prime for each element of the progression. In case of $h=2$ this means that under some plausible unproved hypotheses about regular distribution of the primes and other sequences in arithmetic progressions we really have arbitrarily long arithmetic progressions in the sequence of twin primes.
In this paper, we study the gaps between primes in Beatty sequences following the methods in the recent breakthrough of J. Maynard.
A distinguishing feature of certain intractable problems in prime number theory is the sparsity of the underlying sequence. Motivated by the general problem of finding primes in sparse polynomial sequences, we give an estimate for the number of primes of the shape $x^3 + 2y^3$ where $y$ is small.
We suggest other models of sieve generated sequences like the Sieve of Eratosthenes to explain randomness properties of the prime numbers, like the twin prime conjecture, the lim sup conjecture, the Riemann conjecture, and the prime number theorem.
We consider the second of Mullin's sequences of prime numbers related to Euclid's proof that there are infinitely many primes. We show in particular that it omits infinitely many primes, confirming a conjecture of Cox and van der Poorten.
Properties of Higher-Order Prime Number Sequences
本文研究由素数高阶子序列交错和生成的素数序列的性质，并提出一种通过在实数轴上系统选择与消除素数下标来构造这些序列的新筛法。
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。
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.
Some determinants involving quadratic residues modulo primes
该研究计算了若干模素数二次剩余相关的行列式，证明了一类含正切函数的矩阵行列式在特定素数和整数条件下的精确取值公式，并提出了后续研究的相关猜想。
The distribution of differences of consecutive members of sequences of primes is investigated. A quantitative measure for oscillations among these differences is the curvature of the sequence. If the sequence is not too sparse, then sharp estimates for its curvature are provided.
We prove an upper bound for the least prime in an irrational Beatty sequence. This result may be compared with Linnik's theorem on the least prime in an arithmetic progression.
In this note we describe a method for finding prime numbers as fixed points of particularly simple sequences. Some basic calculations show that success rates for identifying primes this way are over 99.9%. In particular, it seems that the set of odd primes can be obtained as fixed points of the sequence which we call A(1), the sequence of smallest divisors of triangular numbers, where the divisors are positive numbers that have not yet appeared in the sequence.
We consider some quadratic variations of the recursive sequence introduced by Lothar Collatz in 1937.
In this paper, we define a variant of Fibonacci-like sequences that we call prime Fibonacci sequences, where one takes the sum of the previous two terms and returns the smallest odd prime divisor of that sum as the next term. We prove that these sequences always terminate in a power of two but can be extended infinitely to the left.
We will derive a function that eliminates any sequence of equidistant numbers from the integer numbers, then we will derive its inverse. Then we will use the Sequence elimination function to eliminate the multiples of the prime numbers from the integer numbers to produce formulas that generate all prime numbers in some interval.
An Equivalent form of Twin Prime Conjecture connected with a sequence of arithmetic progressions
本文提出了一种与算术进展序列对称性质相关的 twin素数猜想等价形式。
We consider a generalization of Euclid's proof of the infinitude of primes and show that it leads to variants of the Euclid-Mullin sequence that provably contain every prime number.
We show that for any prime prime $p\not=2$ $$\sum_{k=1}^{p-1} {(-1)^k\over k}{-{1\over 2} \choose k} \equiv -\sum_{k=1}^{(p-1)/2}{1\over k} \pmod{p^3}$$ by expressing the l.h.s. as a combination of alternating multiple harmonic sums.
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.
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。
A remark on the distribution of $\sqrt{p}$ modulo one involving primes of special type II
研究特殊类型素数模一的平方根分布问题，证明不等式有无穷多素数解，改进了已有结果。
We consider a particular case of an analog for elliptic curves to the Mersenne problem : finding explicitely all prime power terms in an elliptic divisibility sequence when descent via isogeny is possible. We explain how this question can be related to classical problems in diophantine geometry and we compute an explicit upper bound on the index of prime power terms in magnified elliptic divisibility sequences.
For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $φ(q)$ reduced residue classes modulo $q$. This paper considers patterns of sequences of consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+k})$ modulo $q$. Numerical evidence suggests a preference for certain prime patterns. For example, computed frequencies of the pattern $(a,a)$ modulo $q$ up to $x$ are much less than the expected frequency $π(x)/φ(q)^2$. We begin to rigorously c…
For $n=1,2,3,\ldots$ let $S_n$ be the sum of the first $n$ primes. We mainly show that the sequence $a_n=\root n\of{S_n/n}\ (n=1,2,3,\ldots)$ is strictly decreasing, and moreover the sequence $a_{n+1}/a_n\ (n=10,11,\ldots)$ is strictly increasing. We also formulate similar conjectures involving twin primes or partitions of integers.
In the present paper we investigate distributional properties of sparse sequences modulo almost all prime numbers. We obtain new results for a wide class of sparse sequences which in particular find applications on additive problems and the discrete Littlewood problem related to lower bound estimates of the $L_1$-norm of trigonometric sums.
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.
We consider the primes which divide the denominator of the x-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive prime divisor, one that has not appeared in any earlier term. Proofs of this are known in only a few cases. Weaker results in the general direction are given, using a strong form of Siegel's Theorem and some congruence arguments. Our main result is applied to the study of prime divisors of Somos sequences.
In the present paper we construct normal numbers in base $q$ by concatenating $q$-ary expansions of pseudo polynomials evaluated at the primes. This extends a recent result by Tichy and the author.
Infinite exponential sequences of distinct prime numbers of the form $\lfloor a c^{n^d}+b\rfloor$, $n\geq 0$, are proved to exist for well chosen real constants $a>0$, $b$, $c>1$, $d>1$, assuming Cramer's conjecture on prime gaps. There is an infinity of such prime sequences. Sequences having the least possible growth rate are of particular interest. This work's focus is on prime sequences with $a=1$, $b \in \{0,1\}$, that have the smallest possible constant $c$ given $d>1$, and sequences with the smallest possible $d$, given $c=2$. In particular, we prove the existence of the four infinite ex…
In the present paper the authors construct normal numbers in base $q$ by concatenating $q$-adic expansions of prime powers $\lfloorαp^θ\rfloor$ with $α>0$ and $θ>1$.
We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence $\{F_n\}_{n=0}^\infty$ misses a congruence class modulo every prime large enough. On the other hand, if $Q \neq \pm 1$, we prove that (under GRH) the sequence $\{F_n\}_{n=0}^\infty$ hits every congruence class modulo infinitely many primes.
We use Maynard's methods to show that there are bounded gaps between primes in the sequence $\{\lfloor nα\rfloor\}$, where $α$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties described by Leitmann, we show that for all $m$ there are infinitely many bounded intervals containing $m$ primes and at least one integer of the form $\lfloor f(q)\rfloor$ with $q$ a positive integer.
Let $P(m)$ denote the greatest prime factor of $m$. For integer $a>1$, M. Ram Murty and S. Wong proved that, under the assumption of the ABC conjecture, $$P(a^n-1)\gg_{ε, a} n^{2-ε}$$ for any $ε>0$. We study analogues results for the corresponding divisibility sequence over the function field $\mathbb{F}_q(t)$ and for some divisibility sequences associated to elliptic curves over the rational field $\mathbb{Q}$.
Let $α,β$ be real numbers such that $α>1$ is irrational and of finite type, and let $c$ be a real number in the range $1<c<\frac{14}{13}$. In this paper, it is shown that there are infinitely many Piatetski-Shapiro primes $p = \left\lfloor n^c \right\rfloor$ in the non-homogenous Beatty sequence $\big(\left\lfloorαm+β\right\rfloor\big)_{m=1}^\infty$.
A Diophantine inequality involving different powers of primes of the form $[n^c]$
研究在素数形式[n^c]下，不同幂的素数满足的丢番图不等式，证明存在无限多个素数三元组满足特定不等式条件。
Large prime factors of well-distributed sequences
研究随机元素大素因子的分布，证明当序列具有分布等级1时，大素因子趋向于Poisson-Dirichlet过程，且在正分布等级下，大素因子相关函数趋向于该过程。同时估计最大素因子大于u^{1-ε}的概率为O(ε)。
The main result of the paper is that assuming that the level $θ$ of distribution of primes exceeds 1/2, then there exists a positive $d\leq C(θ)$ such that there are arbitrarily long arithmetic progressions with the property that $p'=p+d$ is the next prime for each element of the progression. If $θ>0.971$, then the above holds for some $d\leq 16$.
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。
Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers
该研究针对带系数互素限制的弗罗贝尼乌斯开放问题，证明此类问题仍存在无法表示的最大整数，并探究其与斯特恩序列等的关联及相关性质。
For integer a let us consider the sequence X_a={x_0,x_1,x_2,...} defined by x_0=a, x_1=1 and, for n>=1, x_{n+1}=x_n+x_{n-1}. We say that a prime p divides X_a if p divides at least one term of the sequence. It is easy to see that every prime p divides X_1, the sequence of Fibonacci numbers. Lagarias, using a technique involving the computation of degrees of various Kummerian extensions first employed by Hasse, showed in 1985 that X_2, the set of primes dividing some Lucas number has natural density 2/3 and posed as a challenge finding the density of prime divisors of X_3. In this paper we reso…
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