Digits of primes

Significant Digits of Primes in Subsets

Benford's Law describes the prevalence of small numbers as the leading digits of numbers in many sets of integers. We prove a variant of Benford's law for many positive-density subsets of the primes. This follows from a more general result over number fields.

Some Notes on Digit Strings in the Primes

Let $S$ be a string of $l$ decimal digits. We give an explicit upper bound on some prime $p$ whose decimal representation contains the string $S$. We also show, as a corollary of the Green-Tao theorem, that there are arbitrarily long arithmetic progressions of prime numbers all of whose decimal representations contain $S$.

On the digital representation of integers with bounded prime factors

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 $b$.

规定素数的二进制数，II

Prescribin the binary digits of the primes, II

我们得到的期望渐近公式的素数$p<N=2^n$与$r$规定（任意放置）的二进制数字，提供$r<cn$为一个合适的常数$c>0$。这个结果改进了我们早先的结果，其中假定$r$满足$r<c(\frac n{\log n})^{4/7}$。

带受限数位的移位素数的Sárközy定理

Sárközy's theorem for shifted primes with restricted digits

该研究针对带受限数位的移位素数，通过构造关联指数和的局部逼近函数证明其具有van der Corput性质，进而得出正上Banach密度的自然数集A中存在满足a₁+p-1=a₂的相关元素及素数p的结论。

An equation about sum of primes with digital sum constraints

We know that any prime number of form $4s+1$ can be written as a sum of two perfect square numbers. As a consequence of Goldbach's weak conjecture, any number great than $10$ can be represented as a sum of four primes. We are motivated to consider an equation with some constraints about digital sum for the four primes. And we conclude that the square root of the digital sum of the four primes will greater than $4$ and will not be a multiple of $3$ if the equation has solutions. In the proof, we give the method of determining whether a number is a perfect square.

Primes and polynomials with restricted digits

Let $q$ be a sufficiently large integer, and $a_0\in\{0,\dots,q-1\}$. We show there are infinitely many prime numbers which do not have the digit $a_0$ in their base $q$ expansion. Similar results are obtained for values of a polynomial (satisfying the necessary local conditions) and if multiple digits are excluded. Our proof is based on the Hardy-Littlewood circle method and Fourier analysis of the set of integers with no digit equal to $a_0$ in base $q$.

The ternary Goldbach problem with two Piatetski-Shapiro primes and a prime with a missing digit

Let $$γ^*=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:.$$ Let $γ^*<γ_0\leq 1$, $c_0=1/γ_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$.\\ We prove on assumption of the Generalized Riemann Hypothesis that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where the $p_i$ are of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$, for $i=1,2$ and the decimal expansion of $p_3$ does not contain the digit $a_0$.\\ The proof merges methods of J. Maynard from his paper on the infinitude of primes with restricted digits, results…

Twenty Digits of Some Integrals of the Prime Zeta Function

The double sum sum_(s >= 1) sum_p 1/(p^s log p^s) = 2.00666645... over the inverse of the product of prime powers p^s and their logarithms, is computed to 24 decimal digits. The sum covers all primes p and all integer exponents s>=1. The calculational strategy is adopted from Cohen's work which basically looks at the fraction as the underivative of the Prime Zeta Function, and then evaluates the integral by numerical methods.

On the distribution of the digits of quotients of integers and primes

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.

规定素数的二进制数

Prescribing the binary digits of primes

我们提出了一个新的结果计数素数$p<N=2^n$，其中$r$（任意放置）数字的二进制扩展$p$指定。与Harman和Katai的早期工作相比，对$r$的限制放宽为$r< c\Big(\frac n{\log n}\Big)^{4/7}$。这个条件的结果从估计Gallagher和Iwaniec的零自由区域的$L$ -功能与“强大的”导体。

沿素数平方的数字函数

Digital functions along Squares of Prime Numbers

本文将Gelfond猜想中沿素数与沿平方的分布问题相结合，证明沿平方的q-可乘函数素数定理，并由此得到沿素数平方的良好分布结果。

$x^{2} + x + 41$形式的60，000位素数

A 60,000 digit prime number of the form $x^{2} + x + 41$

欧拉的观察，多项式$x^{2} + x + 41$采取的$0 \leq x \leq 39$的素数值的动机，我们寻找大值的$x$为$N = x^{2} + x + 41$是素数。为了应用基于$N-1$的因式分解的经典素性证明结果，我们选择$x$具有$g(y)$的形式，选择使得$g(y)^{2} + g(y) + 40$是可约的。我们的主要结果是一个明确的，60,000位素数的形式$x^{2} + x + 41$。

Primes with restricted digits

Let $a_0\in\{0,\dots,9\}$. We show there are infinitely many prime numbers which do not have the digit $a_0$ in their decimal expansion. The proof is an application of the Hardy-Littlewood circle method to a binary problem, and rests on obtaining suitable `Type I' and `Type II' arithmetic information for use in Harman's sieve to control the minor arcs. This is obtained by decorrelating Diophantine conditions which dictate when the Fourier transform of the primes is large from digital conditions which dictate when the Fourier transform of numbers with restricted digits is large. These estimates…

The ternary Goldbach problem with a prime with a missing digit and primes of special types

Let $$γ^*:=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:,\ γ^*<\frac{1}{c_0}\leq 1\:.$$ Let $γ^*<γ_0\leq 1$, $c_0=1/γ_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$. In [23] we proved on assumption of the Generalized Riemann Hypothesis (GRH), that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where for $i=2, 3$ the primes $p_i$ are Piatetski-Shapiro primes - primes of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$ - whereas the decimal expansion of $p_1$ does not contain the digit $a_0$. In this paper we replace …

位置素数数字分布定理：素数中位置数字等分布的形式证明

The Position-wise Prime Digit Distribution Theorem: A Formal Proof of Position-wise Digit Equidistribution in the Prime Numbers

该论文形式证明了素数的位置数字等分布定理，指出内部数字极限概率为1/10、首位数字为1/9，明确了两种机制并给出误差界。

排列和素数

Permutations and primes

讨论了N位数集合的所有排列都是素数的问题。这样的集合可以仅包括数字1、3、7和9，而不包括0、2、5、4、6、8中的任何一个。直接计算表明，这样的全排列数集出现在N = 1，2，3处，而在N的4 - 10区间内不存在。另一方面，在N = 19、23、317和1031（以及N = 2）的情况下，形式上的全置换由代表单位（所有数字都是1的整数）提供。对于任意大的N，其它（非代表）全置换数集的存在/不存在是一个可能有否定答案的公开问题。本文给出了N = 4 - 10时素数个数最多的最大置换数集。

关于有理数的素数展开式的数字模式

On digit patterns in expansions of rational numbers with prime denominator

我们证明了，对于任何固定的$\varepsilon > 0$和几乎所有的素数$p$，任何分数$m/p$与$\gcd(m,p) = 1$的$g$ -ary展开包含几乎所有长度为$k < (5/24 - \varepsilon) \log_g p$的$g$ -ary串.这补充了J. Bourgain，S.科尼亚金和我。E. Shparlinski断言，对于几乎所有的素数，所有长度为$k < (41/504 -\varepsilon) \log_g p$的$g$元串都出现在$m/p$的$g$元展开式中.

缺失一位数字的素数：算术级数中的分布及在筛法理论中的应用

Primes with a missing digit: distribution in arithmetic progressions and an application in sieve theory

本文利用圆法证明了缺失一位数字的素数的Bombieri-Vinogradov型定理，并结合半线性筛，给出了形如p=1+m^2+n^2的此类素数数量的正确数量级上下界。

素数位数和尾分布的界

Bounds For The Tail Distribution Of The Sum Of Digits Of Prime Numbers

设s_q（n）表示以q为基的数字和函数，当n<x时，它以（q-1）/2 log_qx为中心.在Drmota、Mauduit和Rivat 2009年的论文中，他们研究了素数的位数和，并提供了集合{p<x，p prime：s_q（p）=alpha（q-1）log_q x}的大小的渐近性，其中alpha位于1/2附近的严格范围内。在本文中，我们研究了这个分布的尾部，并给出了它的下界|{p < x，p prime：s_q（p）>alpha（q-1）log_q x}|>>x^{2（1-alpha）}e^{-c（log x）^{1/2+ x}}，1/2<alpha<0.7375。为了达到这个下界，我们注意到，多项分布是尖锐的峰值，并适用于短时间间隔的素数的结果。这证明了在二进制展开式中，有无限多个素数的1比0多两倍。

左素截断的分布

Distributions of left prime truncations

研究了整数和多项式左素截断的分布特性，分析了其在所有ℓ位数或多项式中的比例、方差及最大比例。

On the last digits of consecutive primes

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…

Digital images unveil geometric structures in pairs of relatively prime numbers

We present a transformation, based on the Bézout's identity, which maps the set of pairs of relatively prime numbers $(p,q)$ with fixed $p$ and $0<q<p$, to pairs of relatively prime numbers in the $p\times p$ square in $\mathbb R^2$, in such a way that intriguing quadratic arcs show up. We exhibit parametrizations of quadratic curves which fit such quadratic arcs and we also justify algebraically the ensuing geometry.

受限数字素数情形的维诺格拉多夫定理

Vinogradov's theorem for primes with restricted digits

本文在基 g 展开中不含某固定数字的整数集合上，证明了每个充分大的奇数都能表示为三个此类素数之和，从而将维诺格拉多夫三素数定理推广到受限数字的素数情形。

素数数字和的显式满射阈值

An Explicit Surjectivity Threshold for Digit Sums of Primes

基于圆法框架，给出素数数字和满射性的显式数值阈值，并应用于证明OEIS A070027的无穷性。

缺失数字和两个素数平方之和

Missing digits and sums of two prime squares

本文研究了在特定基数下省略固定数字且可表示为两个素数平方之和的整数，利用圆方法和筛法分析了其计数行为并给出了非平凡下界。

On the proximity of large primes

By a sphere-packing argument, we show that there are infinitely many pairs of primes that are close to each other for some metrics on the integers. In particular, for any numeration basis $q$, we show that there are infinitely many pairs of primes the base $q$ expansion of which differ in at most two digits. Likewise, for any fixed integer $t,$ there are infinitely many pairs of primes, the first $t$ digits of which are the same. In another direction, we show that, there is a constant $c$ depending on $q$ such that for infinitely many integers $m$ there are at least $c\log \log m$ primes which…

A note on Benli and Pollack's paper on small prime power residues

In this note, we revisit a result of Benli's and Pollack's on the number of small prime $k^{th}$ power residues. The proof is based on their idea of using reciprocity laws, but the argument is simplified and we prove a slightly stronger bound.

On digits of Mersenne numbers

Motivated by recently developed interest to the distribution of $q$-ary digits of Mersenne numbers $M_p = 2^p-1$, where $p$ is prime, we estimate rational exponential sums with $M_p$, $p \leq X$, modulo a large power of a fixed odd prime $q$. In turn this immediately implies the normality of strings of $q$-ary digits amongst about $(\log X)^{3/2+o(1)}$ rightmost digits of $M_p$, $p \leq X$. Previous results imply this only for about $(\log X)^{1+o(1)}$ rightmost digits.

Chiral Prime Concatenations

The notion of chiral prime concatenations is studied as a recursive construction of prime numbers starting from a seed set and with appropriate blocks to define the primality growth, generation by generation, either from the right or from the left. Several basic questions are addressed like whether chiral concatenation is a symmetric process, an endless process, as well as the calculation of largest chiral prime numbers. In particular, the largest left-concatenated prime number is constructed. It is a unique prime number of 24 digits. By introducing anomalous left-concatenations of primes we c…

关于代数整数幂的β进制展开中排除某一位的β进制展开

On $β$-adic expansions of powers of algebraic integer omitting a digit

本文研究了代数整数幂的β进制展开中排除特定数字的性质，证明了其数量随N的变化规律。

新的数字结果和几个问题

New Digit Results and Several Problems

给出了不同模的Newman数和的一些新关系，并提出了一些问题.特别是，对于奇素数模，我们提出了一个重要的猜想。

缺失数字集中的固定素数准则

A Fixed-Prime Criterion for Reciprocals in Missing-Digit Sets

本文研究缺失数字集K_{m,D}中分数分母的p-adic估值上界，通过固定素数准则推广了Lin等人的工作，并应用于超阶乘、多项式值积和斐波那契数列的倒数。

Leading Digits of Mersenne Numbers

It has long been known that sequences such as the powers of $2$ and the factorials satisfy Benford's Law; that is, leading digits in these sequences occur with frequencies given by $P(d)=\log_{10}(1+1/d)$, $d=1,2,\dots,9$. In this paper, we consider the leading digits of the Mersenne numbers $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime. In light of known irregularities in the distribution of primes, one might expect that the leading digit sequence of $\{M_n\}$ has \emph{worse} distribution properties than "smooth" sequences with similar rates of growth, such as $\{2^{n\log n}\}$. Surprisin…

受限数字差素数

Primes with Restricted-Digit Differences

本文研究差或公差属于受限数字集的素数对及三项算术级数，提出局部傅里叶准则并借助传递算子论证，在显式参数范围内验证了渐近公式的有效性。

两位数定理

Two Digit Theorems

证明了若p是一个具有原根2的素数，则S_p（2 ^p）=p，并给出了S_p（2 ^p）=+或-p类等式的一个充分条件.

Waring's Problem for Polynomial Rings and the Digit Sum of Exponents

Let $F$ be an algebraically closed field of characteristic $p>0$. In this paper we develop methods to represent arbitrary elements of $F[t]$ as sums of perfect $k$-th powers for any $k\in\mathbb{N}$ relatively prime to $p$. Using these methods we establish bounds on the necessary number of $k$-th powers in terms of the sum of the digits of $k$ in its base-$p$ expansion. As one particular application we prove that for any fixed prime $p>2$ and any $ε>0$ the number of $(p^r-1)$-th powers required is $\mathcal{O}\left(r^{(2+ε)\ln(p)}\right)$ as a function of $r$.

Wright's Fourth Prime

Wright proved that there exists a number $c$ such that if $g_0 = c$ and $g_{n+1} = 2^{g_n}$, then $\lfloor g_n \rfloor$ is prime for all $n > 0$. Wright gave $c = 1.9287800$ as an example. This value of $c$ produces three primes, $\lfloor g_1 \rfloor = 3$, $\lfloor g_2 \rfloor = 13$, and $\lfloor g_3 \rfloor = 16381$. But with this $c$, $\lfloor g_4 \rfloor$ is a 4932-digit composite number. However, this slightly larger value of $c$, \[ c = 1.9287800 + 8.2843 \cdot 10^{-4933}, \] reproduces Wright's first three primes and generates a fourth: \[ \lfloor g_4 \rfloor = 19139664204631104984038373…

On $k$-regularity of sequences of valuations and last nonzero digits

Let $b \geq 2$ be an integer base with prime factors $p_1, \ldots, p_s$. In this paper we study sequences of "$b$-adic valuations" and last nonzero digits in $b$-adic expansions of the values $f(n) = (f_1(n), \ldots, f_s(n))$, where each $f_i$ is a $p_i$-adic analytic function. We give a complete classification concerning $k$-regularity of these sequences, which generalizes a result for $b$ prime obtained by Shu and Yao. As an application, we strengthen a theorem by Murru and Sanna on $b$-adic valuations of Lucas sequences of the first kind. Moreover, we derive a method to determine precisely …

素数和Riemann zeta零点的第一位数频率趋于一致，遵循与尺寸相关的广义Benford定律

The first digit frequencies of primes and Riemann zeta zeros tend to uniformity following a size-dependent generalized Benford's law

素数在自然数中的分布似乎除了偶然性外没有其他规律，然而它的全局分布呈现出相当显著的光滑性。随机性和规律性之间的这种相互作用促使各个年龄段的科学家在这种分布中寻找局部和全局模式，最终可以揭示素数的最终性质。在这项工作中，我们表明，一个推广的著名的第一位数本福德定律，它解决了一个给定的领先数字d在数据集中的出现率，描述了惊人的精度领先数字的统计分布在素数序列。此外，这种模式的倒数版本也发生在非平凡的黎曼zeta零点序列中。我们证明，素数定理是，在最后的分析，负责这些模式。还导出了有关素数分布的一些新关系，包括对计数函数pi（n）的一种新的逼近。此外，还导出了与广义本福特定律的统计一致性有关的一些关系式。最后讨论了一些应用。

素数计数 Copeland-Erdős 常数

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。

一个关于Carlitz素幂 torsion 扩张的积分数字导数基

An Integral Digit Derivative Basis for Carlitz Prime Power Torsion Extensions

本文提出了一种用于Carlitz素幂torsion扩张的积分数字导数基，并给出了显式域正交基。

广义数字映射：周期性、Prouhet-Tarry-Escott 解与求和恒等式

A Generalized Digit Map: Periodicity, Prouhet-Tarry-Escott Solutions, and Summation Identities

本文研究基-M到基-N移位映射所得序列b(n)的算术性质，证明其最终周期性条件并界定最小周期，在条件不满足时为Prouhet-Tarry-Escott问题提供新解，并引入有限差分恒等式扩展经典数字和恒等式。

几乎常数可加函数中数字的正规性

The normality of digits in almost constant additive functions

我们考虑由加法函数f（n）中的一些基数b连接而成的数，这些加法函数f（n）非常类似于素计数函数Ω（n）。如果我们将每个f（n）的最后一个\lceil y \frac{\log \log \log n}{\log b} \rceil数字连续连接起来，那么这样创建的数字将是正常的，当且仅当0 < y \le 1/2。这提供了在Erdos-Kac定理失效后可加函数的数字模式的随机性的见解。

On the sum of digits of $1/M$ in $\mathbb{F}_q[x]$

For certain primes $p$, the average digit in the expansion of $1/p$ was found to have a deviation from random behaviour related to the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$ (Girstmair 1994). In this short note, we observe that for the corresponding problem when we replace the integers by polynomials over a finite field, there is never any bias. The argument is elementary.

Primes as sums of Fibonacci numbers

The purpose of this paper is to discuss the relationship between prime numbers and sums of Fibonacci numbers. One of our main results says that for every sufficiently large integer $k$ there exists a prime number that can be represented as the sum of $k$ different and non-consecutive Fibonacci numbers. This property is closely related to, and based on, a prime number theorem for certain morphic sequences. The proof of such a prime number theorem, combined with a corresponding local result, is the central contribution of this paper, from which we derive the result stated in the beginning. Probl…

$d$-平移单位敏感素数

$d$-Translated Unit Sensitive Primes

研究找到对于任意非负整数$d$都是$d$-平移单位敏感的非负整数等差数列，构造出含$k$个连续素数的该数列，找到首个此类已知素数，还优化数列使其元素既是$d$-平移单位敏感的又是布赖尔数且含$k$个连续素数。

Primitive abundant and weird numbers with many prime factors

We give an algorithm to enumerate all primitive abundant numbers (briefly, PANs) with a fixed $Ω$ (the number of prime factors counted with their multiplicity), and explicitly find all PANs up to $Ω=6$, count all PANs and square-free PANs up to $Ω=7$ and count all odd PANs and odd square-free PANs up to $Ω=8$. We find primitive weird numbers (briefly, PWNs) with up to 16 prime factors, improving the previous results of [Amato-Hasler-Melfi-Parton] where PWNs with up to 6 prime factors have been given. The largest PWN we find has 14712 digits: as far as we know, this is the largest example exist…

On large differences between consecutive primes

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.

Arithmetic functions at consecutive shifted 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.

SanD primes and numbers

We define S(um)anD(ifference) numbers as ordered pairs $(m,\, m+Δ)$ such that the digital-sum $DS(m(m+Δ))=Δ.$ We consider both the decimal and the binary case. If both $m$ and $m+Δ$ are prime numbers, we refer to SanD {\em primes}. We show that the number of (decimal-based) SanD numbers less than $x$ grows as $c1\cdot x,$ where $c1 = 2/3,$ while the number of SanD primes less than $x$ grows as $c2\cdot x/\log^2{x},$ where $c2 = 3/4.$ Due to the quasi-fractal nature of the digital-sum function, convergence is both slow and erratic compared to twin primes, which, apart from the constant, have th…

On a conjecture regarding primality of numbers constructed from prepending and appending identical digits

Consider the operation of adding the same number of identical digits to the left and to the right of a number n. In OEIS sequence A090287, it was conjectured that this operation will not produce a prime if and only if n is a palindrome with an even number of digits. We show that this conjecture is false by showing that this property also holds for n=231, n=420, and an infinite number of other values of n.

沿某些数论序列走向无穷

Walking to Infinity Along Some Number Theory sequences

本文研究沿逐项增加一位数字的素数等数论序列走向无穷的猜想，通过贪心与随机模型分析轨迹行为和游走长度，并证明若干序列及其他进制下不可能。

论Telchrid关于等差数列的定理

On Telhcirid's theorem on arithmetic progressions

本文研究反素数（素数的数字逆序）的分布，证明基数足够大时满足简单必要条件的等差数列中反素数有无穷多，还得到反素数的有效Siegel–Walfisz型结果，其容许模水平高于经典情形。