Search arXivSearch

arXiv · 2009.12305

Chiral Prime Concatenations

Abstract

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 can surpass the limit of 24 digits for left-concatenated primes. It is conjectured that prime numbers are left chiral under anomalous concatenations.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Miguel A. Martin-Delgado. 2020-09-25. Chiral Prime Concatenations. https://arxiv.org/abs/2009.12305

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Asymptotic density of k-almost primes

Landau's well known asymptotic formula $$N_k(x):=\ \mid\{n\leq x : Ω(n)=k\}\mid \ \sim \left( \frac{x}{\log x} \right) \frac{(\log\log x)^{k-1}}{(k - 1)!}\ \ (x \rightarrow \infty),$$ which also holds for $$π_k(x):=\ \mid\{n\leq x : ω(n)=k\}\mid,$$ is known to be fairly poor for $k > 1$, and when $k$ is allowed to tend to infinity with $x$, the study of $N_k(x)$ and $π_k(x)$ becomes very technical [1, Chapter II.6, $§$ 6.1, p.200]. I hope to show that the method described below provides not only a more accurate approach, but rather increases in its asymptotic accuracy as $k$ tends to infinity.

math.NT

Transcendence Meets Normality: Construction of Transcendentally Normal Numbers

In this work, we study real numbers $x$ for which $p(x)$ is (absolutely) normal for every non-constant integer-valued polynomial $p$. We call such numbers transcendentally normal. We prove that almost every real number is transcendentally normal and provide an explicit construction of such a number, based on Sierpinski's covering method and novel ideas involving the so-called stretch function. In the next step, we transform this construction into an algorithm that computes the digits of a t-normal number recursively in all integer bases. Moreover, we extend our covering approach to construct and compute LIL-normal numbers whose discrepancies are of the order predicted by the law of the iterated logarithm. We also take the opportunity to discuss several interesting open problems.

math.NT