Simon Plouffe
By using an asymptotic formula known for the numbers of Euler and Bernoulli it is possible to obtain an explicit expression of the nth digit of $π$ in decimal or in binary, it also makes it possible to obtain the $n^{\rm th}$ digit of powers of $π^n$.
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The mathematical constant pi has recently been computed up to 22,459,157,718,361 decimal and 18,651,926,753,033 hexadecimal digits. As a simple check for the normality of pi, the frequencies of all sequences with length one, two and three in the base 10 and base 16 representations are extracted. All evaluated frequencies are found to be consistent with the hypothesis of pi being a normal number in these bases.
On the decimal and octal digits of $1/p$
研究1/p的十进制周期长度及各数字出现频率，基于数论中的二次域类数，探讨十进制和八进制数字分布特性。
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 $S$.
A method for computing the n'th decimal digit of pi in O(n^3 log(n)^3) time and with very little memory is presented here. The computation is based on the recently discovered Bailey-Borwein-Plouffe algorithm and the use of a new algorithm that simply splits an ordinary fraction into its components. The algorithm can be used to compute other numbers like zeta(3), pi*sqrt(3), pi^2 and 2/sqrt(5) ln(phi) where phi is the golden ratio. The computation can be achieved without having to compute the preceding digits. I claim that the algorithm has a more theoretical rather than practical interest, I h…
Completing Loo-Keng Hua's approach to the real number system pioneered in 1962, this paper defines arithmetical operations directly on infinite decimals without appealing to any ordering structure. Therefore, the widespread belief that there exists an algorithm for determining the digits of the product of two real numbers in terms of finite pieces of their decimal strings is essentially confirmed.
We show that for any fixed base $a$, a positive proportion of primes have the property that they become composite after altering any one of their digits in the base $a$ expansion; the case $a=2$ was already established by Cohen-Selfridge and Sun, using some covering congruence ideas of Erdős. Our method is slightly different, using a partially covering set of congruences followed by an application of the Selberg sieve upper bound. As a consequence, it is not always possible to test whether a number is prime from its base $a$ expansion without reading all of its digits. We also present some sli…
It is well known since A. J. Kempner's work that the series of the reciprocals of the positive integers whose the decimal representation does not contain any digit 9, is convergent. This result was extended by F. Irwin and others to deal with the series of the reciprocals of the positive integers whose the decimal representation contains only a limited quantity of each digit of a given nonempty set of digits. Actually, such series are known to be all convergent. Here, letting $S^{(r)}$ $(r \in \mathbb{N})$ denote the series of the reciprocal of the positive integers whose the decimal represent…
Digits of pi: limits to the seeming randomness II
本文通过对12个数学常数（含π的100亿位数字）的十进制表示进行统计分析，发现其数字在中心极限定理和重对数律层面未表现出预期的随机性。
A Canonical Bijection Between Finite-Decimal Real Numbers and Natural Numbers with Constant-Time Enumeration Formulas
本文提出有限小数实数与自然数间的显式双射，通过4元组参数化和闭式公式实现O(1)常数时间的双向映射，全程使用精确十进制算术，证明了有限小数实数可被最优且系统地枚举和索引。
The Prime Digit Distribution Conjecture: A Formal Proof of Average Digit Equidistribution in the Prime Numbers
研究素数数字分布，结合素数定理等方法证明素数中数字出现概率渐近为\(\frac{1}{10}\)，建立内部数字定量等分布，阐明与其他相关问题区别，得出素数数字汇总后渐近等分布的结论。
If we form a decimal where the nth digit is the last non-zero digit of $n!$ (likewise, the last non-zero digit of $n^n$), we obtain an irrational number
We prove that the algorithm of [13] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set $E_2$ (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 deci…
The decimal digits of $π$ are widely believed to behave like as statistically independent random variables taking the values $0, 1, 2, 3, 4, 5$, $6, 7, 8, 9$ with equal probabilities $1/10$. In this article, first, another similar conjecture is explored - the seemingly almost random behaviour of digits in the base 3 representations of powers $2^n$. This conjecture seems to confirm well - it passes even the tests inspired by the Central Limit Theorem and the Law of the Iterated Logarithm. After this, a similar testing of the sequences of digits in the decimal representations of the numbers $π$,…
We first give a summary of the history of transcendental numbers then use a nice technique by G. Dresden to prove a new transcendental number. In particular, while previous work looked at the last non-zero digit of $n^n$, we consider the digit right before its last non-zero digit and show that the infinite decimal built from these digits is transcendental.
The Multiplicative Persistence Conjecture: Resolving the \(2\)-Adic Obstruction for Nonzero Even Targets
本文证明了乘法持久性猜想中关于非零偶目标的2进指数一致有界的部分，得到显式界并推广至任意基，结合已有方法得到分析非零偶终端数字的有限算法。
Numerical verification of the Collatz conjecture for billion digit random numbers
本研究针对数论中著名的未解决问题考拉兹猜想，提出一种可在标准PC上运行的验证算法，实现了对最高100亿位十进制大数的猜想数值验证，拓展了大数验证的可行性。
A Trott number is a number $x\in(0,1)$ whose continued fraction expansion is equal to its base $b$ expansion for a given base $b$, in the following sense: If $x=[0;a_1,a_2,\dots]$, then $x=(0.\hat{a}_1\hat{a}_2\dots)_b$, where $\hat{a}_i$ is the string of digits resulting from writing $a_i$ in base $b$. In this paper we characterize the set of bases for which Trott numbers exist, and show that for these bases, the set $T_b$ of Trott numbers is a complete $G_δ$ set. We prove moreover that the union $T:=\bigcup_{b\geq 2} T_b$ is nowhere dense and has Hausdorff dimension less than one. Finally, w…
Triangular Numbers With a Single Repeated Digit
本文指出Ballew和Weger关于具有单一重复数字的三角形数列表的证明存在缺陷，并提供了一个新的初等证明来纠正该错误。
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.
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…
On the digits of the sum of proper divisors
研究整数\(n\)真因数之和\(s(n)\)数字的概率问题，证明其服从本福特定律，表明多数\(n\)的\(s(n)\)前后\(k(x)\)个数字含所有十进制数字，给出合数\(n\)中\(s(n)\)缺数字情况的上界，揭示素数对结果的影响。
The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when we choose a digit set that does not contain 0. We prove that such digit sets exist and we provide infinitely many examples for every base b with |b|\ge 4, and for b=-2. For the special case b=-2, we give a full characterisation of all valid digit sets.
Abelian-normal decimal expansions
本文引入阿贝尔-正则数的概念，构造了一个非正常类比Champernowne常数$D_{10}$，并证明其在特定加权函数下的阿贝尔-正则性，同时提出两个开放性问题。
A Keith number is a positive integer N with the decimal representation a_1a_2...a_n such that n>=2 and N appears in the sequence (K_m) given by the recurrence K_1=a_1,...,K_n=a_n and K_m=K_{m-1}+K_{m-2}+...+K_{m-n} for m>n. We prove that there are only finitely many Keith numbers using only one decimal digit (i.e., a_1=a_2=...=a_n), and that the set of Keith numbers is of asymptotic density zero.
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…
We give explicit numerical values with 100 decimal digits for the constant in the Mertens product over primes in the arithmetic progressions $a \bmod q$, for $q \in \{3$, ..., $100\}$ and $(a, q) = 1$.
Upper Bounds for Digitwise Generating Functions of Powers of Two: A Problem and a Matrix Representation
本文研究 2^n 十进制展开相关生成函数的渐近上界问题，并给出对应有限状态转移算子的矩阵表示以辅助分析。
In 1973, Neil Sloane published a very short paper introducing an intriguing problem: Pick a decimal integer $n$ and multiply all its digits by each other. Repeat the process until a single digit $Δ(n)$ is obtained. $Δ(n)$ is called the \textsl{multiplicative digital root of $n$} or \textsl{the target of $n$}. The number of steps $Ξ(n)$ needed to reach $Δ(n)$, called the multiplicative persistence of $n$ or \textsl{the height of $n$} is conjectured to always be at most $11$. Like many other very simple to state number-theoretic conjectures, the multiplicative persistence mystery resisted numero…
In this paper we study the $b$-ary expansions of the square roots of the function defined by the recurrence $f_b(n)=b f_b(n-1)+n$ with initial value $f(0)=0$ taken at odd positive integers $n$, of which the special case $b=10$ is often referred to as the "schizophrenic" or "mock-rational" numbers. Defined by Darling in $2004$ and studied in more detail by Brown in $2009$, these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases $…
In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties for fast arithmetic. It is usually redundant, which means that each integer can have multiple different digit expansions, so the natural question for the number of representations arises. In this paper, we provide a general asymptotic formula for the number of such multi-base representations of a positive integer $n$. Moreover, we prove central limit theorems …
Notes on the Finiteness of Powers of Two with All Even Digits
研究2的幂全偶数数字的有限性问题，通过计算验证与基于动态Borel-Cantelli引理的度量论证，证明生成无限此类幂的初始相位集合Lebesgue测度为零，为有限性猜想提供概率支持。
Expansions that furnish increasingly good approximations to real numbers are usually related to dynamical systems. Although comparing dynamical systems seems difficult in general, Lochs was able in 1964 to relate the relative speed of approximation of decimal and regular continued fraction expansions (almost everywhere) to the quotient of the entropies of their dynamical systems. He used detailed knowledge of the continued fraction operator. In 2001, a generalization of Lochs' result was given by Dajani and Fieldsteel in \citeDajF, describing the rate at which the digits of one number-theoreti…
Congruence speed of tetration bases ending with $0$
本文针对以$0$结尾（即$10$的倍数）的迭代幂次基，推导出其在任意给定超指数高度下的同余速度公式，填补了该特定情形下同余速度计算的空白。
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。
In this paper we compare several properties and constructions of the Carlitz polynomials and digit derivatives for continuous functions on $\F_q[[T]].$ In particular, we show a close relation between them as orthonormal bases. Moreover, parallel to Carlitz's coefficient formula, we give the closed formula for the expansion coefficients in terms of the digit derivatives.
BBP-type formulas are usually discovered experimentally, one at a time and in specific bases, through computer searches. In this paper, however, we derive directly, without doing any searches, explicit digit extraction BBP-type formulas in general binary bases $b=2^{12p}$, for $p$ positive odd integers. As particular examples, new binary formulas are presented for $π\sqrt 3$, $π\sqrt 3\log 2$, $\sqrt 3\;{\rm Cl}_2(π/3)$ and a couple of other polylogarithm constants. A variant of the formula for $π\sqrt 3\log 2$ derived in this paper has been known for over ten years but was hitherto unproved. …
Cryptarithmically unique terms in integer sequences
本文研究整数序列中具有唯一数字模式的密码算术唯一项，估计其在平方数和素数中的密度，并引入素数不具备的恐素数数字模式。
This article gives a direct formula for the computation of B(n) using the asymptotic formula $$B (n) \approx 2 {\frac {n!}{π^{n}{2}^{n}}}$$ where n is even and $n >> 1$. This is simply based on the fact that $ζ(n)$ is very near 1 when n is large and since $B (n) = 2 {\frac {ζ(n) n!}{π^{n}{2}^{n}}}$ exactly. The formula chosen for the Zeta function is the one with prime numbers from the well-known Euler product for $ζ(n)$. This algorithm is far better than the recurrence formula for the Bernoulli numbers even if each B(n) is computed individually. The author could compute $B (750,000)$ in a few…
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