相关论文: Digit Reversal Without Apology

The first digit law, also known as Benford's law or the significant digit law, is an empirical phenomenon that the leading digit of numbers from real world sources favors small ones in a form $\log(1+{1}/{d})$, where $d=1, 2, ..., 9$. Such…

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…

We prove Bombieri-Vinogradov type theorems for primes with a missing digit in their $b$-adic expansion for some large positive integer $b$. The proof is based on the circle method, which relies on the Fourier structure of the integers with…

We prove a conjecture posted in the Online Encyclopedia of Integer Sequences, namely that there are exactly five positive integers that can be written in more than one way as the sum of a nonnegative power of 2 and a nonnegative power of 3.…

The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation. We also…

The well-known 1089 trick reflects an amazing trait of digital reversal process and reminisces of a limiting attractor in dynamical systems even though it takes only two steps. It is natural to consider the situations when the number of…

A permutiple is the product of a digit preserving multiplication, that is, a number which is an integer multiple of some permutation of its digits. Certain permutiple problems, particularly transposable, cyclic, and, more recently,…

The problem of N-digit sets all permutations of which give primes is discussed. Such sets may include only digits 1, 3, 7 and 9, and none of 0, 2, 5, 4, 6, 8. Direct calculations show that such full-permutation digit sets occur at N = 1, 2,…

A. V. Arhangel'ski\u{i} introduced in 2012, when he was visiting the department of Mathematics at King Abduaziz University, new weaker versions of normality, called \it $C$-normality, \rm and \it countable normality. \rm The purpose of this…

The uneven distribution of digits in numerical data, known as Benford's law, was discovered in 1881. Since then, this law has been shown to be correct in copious numerical data relating to economics, physics and even prime numbers. Although…

We developed an algorithm that easily goes from one odd number to the next odd number in binary representation for the reduced forward Collatz map (Syracuse function). The algorithm indicates when an odd number can grow or shrink to the…

In the note, the authors give a unified proof of Identities~67, 84, and~85 in the monograph "M. Z. Spivey, The Art of Proving Binomial Identities, Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 2019; available online…

Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0, 1, ..., s-1 occurs with the same frequency 1/s. Let X be the set of positive integers that are not perfect…

There is one, and only one way, consistent with fundamental physics, that the efficiency of general digital computation can continue increasing indefinitely, and that is to apply the principles of reversible computing. We need to begin…

We show that, for any $r\geq 1$, if $g_1,\ldots,g_r$ are distinct coprime integers, sufficiently large depending only on $r$, then for any $\epsilon>0$ there are infinitely many integers $n$ such that all but $\epsilon \log n$ of the digits…

Set out here are some fundamental theories that may be regarded as newly discovered metamathematics of the odd integers in relation to the Collatz conjecture (also called the 3x+1 problem). Originally motivated by the requirement to invent…

Let $ \{P_{n}\}_{n\geq 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 =1=P_2$ and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\geq 0 $. In this paper, we find all repdigits in base $ 10 $ which can be written as a sum of three…

A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with convergent…

This problem was asked to K. Mahler by one of his Japanese colleagues, a Z-number is a positive real number $x$ such that the fractional parts of $x(\frac{3}{2})^n $ are less than $\frac{1}{2}$ for all integers $n$ such that $n \ge 0$. Kurt…

The decimal digits of $\pi$ 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…