Related papers: Analysis of Prime Reciprocal Sequences in Base 10
The analysis of the decoding failure rate of the bit-flipping algorithm has received increasing attention. For a binary linear code we consider the minimum number of rows in a parity-check matrix such that the bit-flipping algorithm is able…
This paper analyzes the emergence and distribution of potential twin primes, pairs of integers that are both relatively prime to the first n primes or to a given set M of primes, and which are the breeding grounds of true twin primes. It…
Similarly to their purely electric counterparts, spintronic circuits may be presented as networks of lumped elements. Due to interplay between spin and charge currents, each element is described by a matrix conductance. We establish…
We study pairs of consecutive odd numbers through a straightforward indexing. We focus in particular on twin primes and their distribution. With a counting argument, we calculate the limit of an alternating sum that is equal to 1 which…
We propose two new kinds of infinite resistor networks based on the Fibonacci sequence: a serial association of resistor sets connected in parallel (type 1) or a parallel association of resistor sets connected in series (type 2). We show…
In this article, we try to explain and unify standard divisibility tests found in various books. We then look at recurring decimals, and list a few of their properties. We show how to compute the number of digits in the recurring part of…
We determine the minimal spectral radii among all skew-reciprocal integer matrices of a fixed even dimension that are primitive or nonnegative and irreducible. In particular, except for dimension six, we show that each such class of…
In a recent work Friedlander studied the problem of how large consecutive prime gaps should be in order that the sum of the reciprocals should be divergent. Supposing a very deep Hypothesis, a generalization of the Hardy--Littlewood prime…
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…
We give heuristic arguments and computer results to support the hypothesis that, after appropriate rescaling, the statistics of spacings between adjacent prime numbers follows the Poisson distribution. The scaling transformation removes the…
The partial sums of integer sequences that count the occurrences of a specific pattern in the binary expansion of positive integers have been investigated by different authors since the 1950s. In this note, we introduce generalized pattern…
We study, from the viewpoint of metrical number theory and (infinite) ergodic theory, the probabilistic laws governing the occurrence of prime numbers as digits in continued fraction expansions of real numbers.
We study the effects of finite-precision representation of source's probabilities on the efficiency of classic source coding algorithms, such as Shannon, Gilbert-Moore, or arithmetic codes. In particular, we establish the following simple…
Mapping between sequence and structure is currently an open problem in structural biology. Despite many experimental and computational efforts it is not clear yet how the structure is encoded in the sequence. Answering this question may…
Adversarial examples pose a unique challenge for deep learning systems. Despite recent advances in both attacks and defenses, there is still a lack of clarity and consensus in the community about the true nature and underlying properties of…
The results for the fractional sequence $\left \{[x/n]+1:n \leq x\right \}$, and the fractional sequence in arithmetic progression $\left \{q[x/n]+a:n \leq x\right \}$, where $a<q$ are integers such that $\gcd(a,q)=1$, prove that these…
The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…
In the course of studies of the measure of chaos for the distribution of the prime numbers among the positive integers N arched structures have been found. It is given a brief description of the fine structure of the positive integers…
Primes in the two complete associative normed division algebras C and H have affinities with structures seen in the standard model of particle physics. On the integers in the two algebras, there are two equivalence relations: a strong one,…
In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this…