Related papers: The gaps in the multiplication table
An overview of the results of new exhaustive computations of gaps between primes in arithmetic progressions is presented. We also give new numerical results for exceptionally large least primes in arithmetic progressions.
For a prime number $p$, we consider its primorial $P:=p\#$ and $U(P):={\left(\mathbb{Z}/P\mathbb{Z}\right)}^\times$ the set of elements of the multiplicative group of integers modulo $P$ which we represent as points anticlockwise on a…
We characterise gaps in the full homomorphism order of graphs.
We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations.…
We consider the notion of multiple gap as a finite set of ideals that cannot be separated. We study the different types of such objects that can be found in the Boolean algebra of subsets of the natural numbers modulo finite sets.
For any positive integer $k$, we show that infinitely often, perfect $k$-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$ c_k \frac{\log p \log_2 p \log_4 p}{(\log_3 p)^2}, $$ where $p$ is…
We show that a positive proportion of all gaps between consecutive primes are small gaps. We provide several quantitative results, some unconditional and some conditional, in this flavour.
In this paper, we show some results about the gap between a prime number and its consecutive prime number for large enough prime numbers. We show that the gap between a prime number $p_n$ and its consecutive prime number is not larger than…
We study whether several consecutive prime gaps can all be relatively large at the same time, or is it possible that all are squares or perfect powers, or perhaps none of them are squares? A few related results and problems are also…
We explicitly describe all the isolated gaps of any numerical semigroup of embedding dimension two, and we give an exact formula for the number of isolated gaps of these numerical semigroups.
We identify the finite list of minimal analytic n-gaps which are not countably separated, and we prove that every analytic n-gap which is not countably separated contains a gap from our finite list.
We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $\pi(x)$. As the application of this formula we formulate 7…
This is an introductory article to the theory of multiple gaps.
In this paper, we show a new upper bound of prime gaps, that is the gap between a prime number and its consecutive prime number. We show that the gap between a prime number $p_n$ and its consecutive prime number is not larger than…
We present tables of record (maximal) gaps between densest prime constellations, or k-tuplets. The tables contain all maximal gaps between prime k-tuplets up to 10^15, for each k<=7.
Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…
Gap balancing numbers are a certain generalization of balancing and cobalancing numbers that arise from studying the equation ${T(L)+T(B)=T(m)}$ where $T(i)$ is the $i$th triangular number. In this paper, we survey early results, attempt to…
In this article, a relation between a gap $d_{k}$ and divisors of composite numbers between $p_{k}$ and $p_{k+1}$ is established.
We study the problem of estimating the number of points of coincidences of an idealized gap on the set of integers under a given multiplicative function $g:\mathbb{N}\longrightarrow \mathbb{C}$ respectively additive function…
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…