Kota Saito
In 2010, Matomäki investigated the set of $A>1$ such that the integer part of $ A^{c^k} $ is a prime number for every $k\in \mathbb{N}$, where $c\geq 2$ is any fixed real number. She proved that the set is uncountable, nowhere dense, and has Lebesgue measure $0$. In this article, we show that the set has Hausdorff dimension $1$.
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It is well known that the arithmetic nature of Mills' prime-representing constant is uncertain: we do not know if Mills' constant is a rational or irrational number. In the case of other prime-representing constants, irrationality can be proved, but it is not known whether these constants are algebraic or transcendental numbers. By using Liouville or Roth's theorems about approximation by rationals, we find a couple of prime-representing constants that can be proved to be transcendental numbers.
We present an elementary inductive proof which Euler could have obtained, for the corresponding result as the title indicates, had he refined a bit his proof for Fermat's assertion on representing primes as two squares.
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 present a constant and a recursive relation to define a sequence $f_n$ such that the floor of $f_n$ is the $n$th prime. Therefore, this constant generates the complete sequence of primes. We also show this constant is irrational and consider other sequences that can be generated using the same method.
Exact dimensions of the prime continued fraction Cantor set
本文研究了素数Cantor集的Hausdorff维数和packing维数，证明了其Hausdorff测度不能有限正，但在数论猜想下其packing测度关于特定维数函数正且有限。
Mills proved that there exists a real constant $A>1$ such that for all $n\in \mathbb{N}$ the values $\lfloor A^{3^n}\rfloor$ are prime numbers. No explicit value of $A$ is known, but assuming the Riemann hypothesis one can choose $A= 1.3063778838\ldots .$ Here we give a first unconditional variant: $\lfloor A^{10^{10n}}\rfloor$ is prime, where $A=1.00536773279814724017\ldots$ can be computed to millions of digits. Similarly, $\lfloor A^{3^{13n}}\rfloor$ is prime, with $A=3.8249998073439146171615551375\ldots .$
We give criteria of the solvability of the diophantine equation $p=x^2+ny^2$ over some imaginary quadratic fields where $p$ is a prime element. The criteria becomes quite simple in special cases.
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。
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.
Mills showed that there exists a constant $A$ such that $\lfloor{A^{3^n}}\rfloor$ is prime for every positive integer $n$. Kuipers and Ansari generalized this result to $\lfloor{A^{c^n}}\rfloor$ where $c\in\mathbb{R}$ and $c\geq 2.106$. The main contribution of this paper is a proof that the function $\lceil{B^{c^n}}\rceil$ is also a prime-representing function, where $\lceil X\rceil$ denotes the ceiling or least integer function. Moreover, the first 10 primes in the sequence generated in the case $c=3$ are calculated. Lastly, the value of $B$ is approximated to the first $5500$ digits and is …
We prove that, for every modulus $\mathfrak{q}$, every class of the narrow ray class group $H_{\mathfrak{q}}(\mathbf{K})$ of an arbitrary number field $\mathbf{K}$ contains a product of three unramified prime ideals $\mathfrak{p}$ of degree one with $\mathfrak{N}\mathfrak{p}\le (t(\mathbf{K})\mathfrak{N}\mathfrak{q})^3$, where $t(\mathbf{K})$ is an explicit function of $\mathbf{K}$ described in the paper. To achieve this result, we first obtain a sharp explicit Brun-Titchmarsh Theorem for ray classes and then an equally explicit improved Brun-Titchmarsh Theorem for large subgroups of narrow ra…
We prove a version of Garaev's sum product theorem in the set of finite fields with non-prime order. Because of the presence of subfields, this seems to require some hypotheses on the set. We work under a condition analogous to having Hausdorff dimension less than 1/2. Under these conditions, we obtain a sum-product theorem with exponent 49/48.
Intrinsic Diophantine approximation by rationals of height with a bounded number of distinct prime factors
本文研究了由具有有限不同素因子数的有理数逼近的自相似集的Hausdorff维数，修正了前一版本中引理3.8的证明错误。
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.
Let $(c_k)_{k\in \mathbb{N}}$ be a sequence of positive integers. We investigate the set of $A>1$ such that the integer part of $A^{c_1\cdots c_k}$ is always a prime number for every positive integer $k$. Let $\mathcal{W}(c_k)$ be this set. The first goal of this article is to determine the topological structure of $\mathcal{W}(c_k)$. Under some conditions on $(c_k)_{k\in \mathbb{N}}$, we reveal that $\mathcal{W}(c_k)\cap [0,a]$ is homeomorphic to the Cantor middle third set for some $a$. The second goal is to propose an algebraically independent subset of $\mathcal{W}(c_k)$ if $c_k$ is rapidl…
Let P be a finite set of at least two prime numbers, and A the set of positive integers that are products of powers of primes from P. Let F(k) denote the smallest positive integer which cannot be presented as sum of less than k terms of A. In a recent paper Nathanson asked to determine the properties of the function F(k), in particular to estimate its growth rate. In this paper we derive several results on F(k) and on the related function which denotes the smallest positive integer which cannot be presented as sum of less than k terms from the union of A and -A.
Continued fractions with large prime partial quotients
本文研究部分商为大素数的连分数实数集合，通过证明几乎素数zeta函数尾端的新渐近结果，确定了相应集合的勒贝格测度与豪斯多夫维数，推广并涵盖了多个已有经典与近期定理。
A classical theorem in number theory due to Euler states that a positive integer $z$ can be written as the sum of two squares if and only if all prime factors $q$ of $z$, with $q\equiv 3 \pmod{4}$, have even exponent in the prime factorization of $z$. One can consider a minor variation of this theorem by not allowing the use of zero as a summand in the representation of $z$ as the sum of two squares. Viewing each of these questions in $\Z_n$, the ring of integers modulo $n$, we give a characterization of all integers $n\ge 2$ such that every $z\in \Z_n$ can be written as the sum of two squares…
In this paper, we show that if $p\equiv 1\pmod 4$ is prime, then $4F_p$ admits a representation of the form $u^2-pv^2$ for some integers $u$ and $v$, where $F_n$ is the $n$th Fibonacci number. We prove a similar result when $p\equiv -1\pmod 4$.
Rational dynamics of a prime-representing map
本文研究了素数表示常数递归构造中出现的映射$\mathcal{T}(x)=\lfloor x floor(1+\{x\})$的有理动力学，定义了有理数的阶，并证明了有限阶有理数的自然密度为1，同时给出了特定分母下的显式分数家族。
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.
We consider the exceptional set in the binary Goldbach problem for sums of two almost twin primes. Our main result is a power-saving bound for the exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at most $2$ prime divisors and $p_2+2$ has at most $3$ prime divisors. There are three main ingredients in the proof: a new transference principle like approach for sieves, a combination of the level of distribution estimates of Bombieri--Friedlander--Iwaniec and Maynard with ideas of Drappeau to produce power savings, and a generalisation of the circle method arguments of …
Let $Λ(n)$ be the von Mangoldt function, $x$ real and $2\leq y \leq x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ S_k(x,y;α) = \sum_{x< n \leq x+y} Λ(n) e\left( n^k α\right) \] when $k\geq 3$ for $α$ in the minor arcs. And then combined with the Hardy--Littlewood circle method, this enables us to investigate the Waring--Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers, which improves the results in Wei and Wooley [12].
In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X^2+nY^2. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n=2. In this paper, we prove that in fact this constant is unbounded as n runs through positive integers with a fixed number of prime divisors.
Recursive Prime Factorizations: Dyck Words as Numbers
本文提出了一种基于Dyck词的非位置数制系统，通过递归扩展素因数分解来唯一表示自然数和有理数，并探讨了Dyck完备语言的可能研究方向。
A sequence of integers $ \{ s_n \}_{n \in \mathbb{N}} $ is called a T-sequence if there exists a Hausdorff group topology on $ \mathbb{Z} $ such that $ \{ s_n \}_{n \in \mathbb{N}} $ converges to zero. For every finite set of primes $ S $ we build a Hausdorff group topology on $ \mathbb{Z} $ such that every growing sequence of $ S $-integers converges to zero. As a corollary, we solve in the affirmative an open problem by I.V. Protasov and E.G. Zelenuk asking if $ \{ 2^n + 3^n \}_{n \in \mathbb{N}} $ is a T-sequence. Our results rely on a nontrivial number-theoretic fact about $ S $-unit equat…
Positive Logarithmic Hausdorff Measures of Exceptional Sets for the $p$-adic and $t$-adic Littlewood Conjectures
该研究证明p进Littlewood猜想例外集若非空则具正对数Hausdorff测度，还得到有限域上t进情形的更强结论，改进了Lai和Sprang的构造，特征2情形的结论仍待反例存在性验证。
On arithmetic terms expressing the prime-counting function and the n-th prime
本研究提出首个固定长度的初等闭式算术项，可分别表示素数计数函数$\pi(n)$与第$n$个素数$p(n)$，仅用有限固定次数的初等算术运算，为素数是否存在次序的基础问题提供构造性解答。
Counting and Hausdorff measures for integers and $p$-adic integers
本文利用 $p$-进整数的分形结构研究整数集合，建立计数测度界、计数维数与盒计数维数关系，并给出整数集为闭集投影的充要组合条件。
Transcendence of Montgomery Reduction Factor in the Ring of Integers Modulo Infinitely Large Primes
本文证明在模无穷大素数整数环中，Montgomery 归约因子 $R'$ 的像在有理数域上具有超越性，其中 $R'$ 定义为满足特定不等式的 $2$ 的幂的模逆。
Gaussian rational numbers in Cantor sets in the complex plane
本文研究了复平面上康托集的高斯有理数分布，证明当Hausdorff维数小于1时，此类数的分母素因子有限。
Let $q$ be a perfect power of a prime number $p$ and $E({\mathbb F}_q)$ be an elliptic curve over ${\mathbb F}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer $n$ we denote by $ \# E({\mathbb F}_{q^n})$ the number of rational points on $E$ (including infinity) over the extension ${\mathbb F}_{q^n}$. Under a mild technical condition, we show that the sequence $\lbrace \# E({\mathbb F}_{q^n}) \rbrace_{n>0}$ contains at most $10^{200}$ perfect squares. If the mild condition is not satisfied, then $\#E({\mathbb F}_{q^n})$ is a perfect square for infinitely many $n$ including all th…
Arithmetic limiting laws over shifted primes: dimension zero, Fourier decay, and absolute continuity
研究素数域上矩阵行列式为原根的比例分布，证明其极限分布具有Hausdorff维数为零的特性，并探讨其对PRIM-LWE在密码学中的影响。
Disproof of the uniform Littlewood conjecture
本文通过构造性方法证明了统一利特伍德猜想不成立，展示了反例集具有正豪斯多夫维数，并反驳了p进数利特伍德问题的统一版本及更一般S-算术设置中的弱化版本。
Irreducibility criteria for pairs of polynomials whose resultant is a prime number
本文针对结式为素数或含大素因子的整数系数多项式对，给出不可约性判据，并推广至二元多项式对，通过结式分解与根距上界不可约因子个数。
Zero-full law for well approximable sets in missing digit sets
本文研究缺数字集中的数被 $t$ 进幂分母有理数逼近的问题，证明多种情形下 Hausdorff 测度的零-满定律，并纠正、推广已有定理及给出维数界。
In this report we present an off-the-number-line representation of the positive integers by expressing each integer by its unique prime signature as a grid point of an infinite dimensional space indexed by the prime numbers, which we term the prime grid. In this space we consider a zigzag line, termed the number trail that starts at the origin (representing 1) and travels through every single grid point in the order of the increasing sequence of the natural numbers. Using the infinity norm we define an arithmetic function $L_\infty (N)$ tabulating the total length of the zigzag up to the integ…
The paper exploits an isomorphism between the natural numbers N and a space U of periodic sequences of the roots of unity in constructing a recursive procedure for representing and computing the prime numbers. The nth wave number ${\bf u}_n$ is the countable sequence of the nth roots of unity having frequencies k/n for all integer phases k. The space U is closed under a commutative and associative binary operation ${\bf u}_m \odot{\bf u}_n={\bf u}_{mn}$, termed the circular product, and is isomorphic with N under their respective product operators. Functions are defined on U that partition wav…
On Sums of Powers of Almost Equal Primes
本文研究几乎相等素数的幂和表示问题，通过新的Weyl型估计，证明了在满足同余条件且素数个数足够多时，大整数可表示为$s$个几乎相等素数的$k$次幂之和。
We consider representing of natural numbers by arithmetical expressions using ones, addition, multiplication and parentheses. The (integer) complexity of n -- denoted by ||n|| -- is defined as the number of ones in the shortest expressions representing n. We arrive here very soon at the problems that are easy to formulate, but (it seems) extremely hard to solve. In this paper we represent our attempts to explore the field by means of experimental mathematics. Having computed the values of ||n|| up to 10^12 we present our observations. One of them (if true) implies that there is an infinite num…
Representing the inverse map as a composition of quadratics in a finite field of characteristic $2$
本文研究特征2有限域上逆映射的二次函数复合表示，扩展Petrides结果并提出数论方法，将可处理的奇数指数范围至少推进到250。
Let $p$ be a prime number. For a positive integer $n$ and a real number $ξ$, let $λ_n (ξ)$ denote the supremum of the real numbers $λ$ for which there are infinitely many integer tuples $(x_0, x_1, \ldots , x_n)$ such that $| x_0 ξ- x_1|_p, \ldots , | x_0 ξ^n - x_n|_p$ are all less than $X^{-λ- 1}$, where $X$ is the maximum of $|x_0|, |x_1|, \ldots , |x_n|$. We establish new results on the Hausdorff dimension of the set of real numbers $ξ$ for which $λ_n (ξ)$ is equal to (or greater than or equal to) a given value.
Machine Learnability as a Measure of Order in Aperiodic Sequences
本文通过图像聚焦的机器学习模型，研究素数分布区域的规律性，发现更高区域的素数分布更易被学习，揭示了机器学习在数论研究中的潜力。
Coprime-Universal Quadratic Forms
给定素数 $p>3$，本文刻画了对 $p$ 是 coprime-universal 的正定整二次型，推广了 $290$ 定理，并在 GRH 下证明了特定情况，最后探讨了等差数列的表示问题。
The Prime Clockwork: A Dynamic Representation of Modular and Multiplicative Arithmetic
该研究提出素数发条这一离散动力系统，无需显式模除等运算即可生成素数并表示整数，其大循环对应中国剩余定理，还可用于计算多种数论函数。
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