Showing papers on "Prime number published in 2016"
TL;DR: A precise conjecture is formulated, based on the Hardy−Littlewood conjectures, which predicts that all patterns do occur their fair share of the time in the limit, but that there are secondary terms only very slowly tending to zero that create the observed biases.
Abstract: Although the sequence of primes is very well distributed in the reduced residue classes [Formula: see text], the distribution of pairs of consecutive primes among the permissible ϕ(q)(2) pairs of reduced residue classes [Formula: see text] is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared with numerical data, and the observed fit is very good.
85 citations
TL;DR: A Mobius-randomness-principle is proved for automatic sequences from which the Sarnak conjecture for this class of sequences is deduced and a Prime Number Theorem is shown forautomatic sequences that are generated by strongly connected automata.
Abstract: We present in this paper a new method to deal with automatic sequences.
This method allows us to prove a Mobius-randomness-principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences.
Furthermore, we can show a Prime Number Theorem for automatic sequences that are generated by strongly connected automata where the initial state is fixed by the transition corresponding to $0$.
48 citations
Book•
1 Apr 2016TL;DR: In this article, the Riemann hypothesis of prime numbers is discussed and the authors provide an accessible explanation of the key ideas of this conjecture, which remains one of the most important unsolved problems in mathematics.
Abstract: Prime numbers are beautiful, mysterious, and beguiling mathematical objects. The mathematician Bernhard Riemann made a celebrated conjecture about primes in 1859, the so-called Riemann hypothesis, which remains one of the most important unsolved problems in mathematics. Through the deep insights of the authors, this book introduces primes and explains the Riemann hypothesis. Students with a minimal mathematical background and scholars alike will enjoy this comprehensive discussion of primes. The first part of the book will inspire the curiosity of a general reader with an accessible explanation of the key ideas. The exposition of these ideas is generously illuminated by computational graphics that exhibit the key concepts and phenomena in enticing detail. Readers with more mathematical experience will then go deeper into the structure of primes and see how the Riemann hypothesis relates to Fourier analysis using the vocabulary of spectra. Readers with a strong mathematical background will be able to connect these ideas to historical formulations of the Riemann hypothesis.
46 citations
TL;DR: In this paper, it was shown that at least 12.5% of all nonnegative real numbers belong to the limit points of the sequence tppn`1 ´ pnq{logpnu 8"1 of normalized differences between consecutive primes.
Abstract: Let pn denote the nth smallest prime number, and let L denote the set of limit points of the sequence tppn`1 ´ pnq{logpnu 8"1 of normalized differences between consecutive primes. We show that for k " 9 and for any sequence of k nonnegative real numbers β1 ď β2 ď ¨ ¨ ¨ ď βk, at least one of the numbers βj ´ βi (1 ď i a j ď k) belongs to L. It follows that at least 12.5% of all nonnegative real numbers belong to L.
43 citations
1 Dec 2016
TL;DR: For the Riemann zeta function, this article showed that there is a prime between $n^3$ and $n+1)^3 for all ε > 0.
Abstract: We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.3))$. This is done by first deriving the Riemann--von Mangoldt explicit formula for the Riemann zeta-function with explicit bounds on the error term. We use this along with other recent explicit estimates regarding the zeroes of the Riemann zeta-function to obtain the result. Furthermore, we show that there is a prime between any two consecutive $m$th powers for $m \geq 5 \times 10^9$. Notably, many of the explicit estimates developed in this paper can also find utility elsewhere in the theory of numbers.
34 citations
TL;DR: For a very general principally polarized complex abelian 3-fold, the Chow group of algebraic cycles is innite modulo every prime number as mentioned in this paper, and this gives the
Abstract: For a very general principally polarized complex abelian 3-fold, the Chow group of algebraic cycles is innite modulo every prime number. In particular, this gives the
27 citations
TL;DR: In this article, it was shown that if a function exhibits polynomial (or Gowers anti-uniform) behaviour in two different directions separately, then it also exhibits the same behavior (but at higher degree) in both directions jointly.
Abstract: We establish a number of "concatenation theorems" that assert, roughly speaking, that if a function exhibits "polynomial" (or "Gowers anti-uniform", "uniformly almost periodic", or "nilsequence") behaviour in two different directions separately, then it also exhibits the same behavior (but at higher degree) in both directions jointly. Among other things, this allows one to control averaged local Gowers uniformity norms by global Gowers uniformity norms. In a sequel to this paper, we will apply such control to obtain asymptotics for "polynomial progressions" $n+P_1(r),\dots,n+P_k(r)$ in various sets of integers, such as the prime numbers.
21 citations
TL;DR: In this paper, it was shown that the variance of the primes not exceeding a real number x obeys Taylor's law asymptotically for large x, where x → ∞.
Abstract: ASBTRACTTaylor's law, which originated in ecology, states that, in sets of measurements of population density, the sample variance is approximately proportional to a power of the sample mean. Taylor's law has been verified for many species ranging from bacterial to human. Here, we show that the variance V(x) and the mean M(x) of the primes not exceeding a real number x obey Taylor's law asymptotically for large x. Specifically, V(x) ∼ (1/3)(M(x))2 as x → ∞. This apparently new fact about primes shows that Taylor's law may arise in the absence of biological processes, and that patterns discovered in biological data can suggest novel questions in number theory. If the Hardy-Littlewood twin primes conjecture is true, then the identical Taylor's law holds also for twin primes. Taylor's law holds in both instances because the primes (and the twin primes, given the conjecture) not exceeding x are asymptotically uniformly distributed on the integers in [2, x]. Hence, asymptotically M(x) ∼ x/2, V(x) ∼ x2/12. High...
18 citations
Posted Content•
TL;DR: Euclid's proof of the infinitude of prime numbers is recast as a Euclidean criterion for a domain to have infinitely many atoms and it is shown that this criterion applies even in certain domains in which not all nonzero nonunits factor into products of irreducibles.
Abstract: We recast Euclid's proof of the infinitude of prime numbers as a Euclidean Criterion for a domain to have infinitely many atoms. We make connections with Furstenberg's "topological" proof of the infinitude of prime numbers and show that our criterion applies even in certain domains in which not all nonzero nonunits factor into products of irreducibles.
17 citations
TL;DR: In this article, it was shown that for every such c there are infinitely many members of P c having at most R (c ) prime factors, giving explicit estimates for R ( c ) when c is near one and also when C is large.
TL;DR: In this paper, a conjecture on supercongruences for the Almkvist-Zudilin numbers was proved for the special case of the superconconcongruence problem.
Abstract: Given a prime number $p$, the study of divisibility properties of a sequence $c(n)$ has two contending approaches: $p$-adic valuations and superconcongruences. The former searches for the highest power of $p$ dividing $c(n)$, for each $n$; while the latter (essentially) focuses on the maximal powers $r$ and $t$ such that $c(p^rn)$ is congruent to $c(p^{r-1}n)$ modulo $p^t$. This is called supercongruence. In this paper, we prove a conjecture on supercongruences for sequences that have come to be known as the Almkvist-Zudilin numbers. Some other (naturally) related family of sequences will be considered in a similar vain.
1 Jul 2016
TL;DR: In this article, the authors prove existence of a set of positive real numbers whose logarithmic measure is finite, such that they can improve the error term of the prime geodesic theorem as $x\to\infty$ $(x
otin E)$.
Abstract: We prove existence of a set $E$ of positive real numbers, which is relatively small in the sense that its logarithmic measure is finite, such that we can improve the error term of the prime geodesic theorem as $x\to\infty$ $(x
otin E)$. The result holds for any compact hyperbolic surfaces, and it would also be true for generic hyperbolic surfaces of finite volume according to the philosophy of Phillips and Sarnak.
TL;DR: In this article, the general structure and properties of Boolean hypercubes are applied to discuss Goldbach's conjecture and a simple reasoning is developed to show that any even unsigned integer, belonging to the interval comprised between two successive powers of two, complies with the fact that it can be expressed as the sum of all the odd integers in the interval from 1 up to the nearest upper power of two.
Abstract: The general structure and properties of Boolean Hypercubes is applied to discuss Goldbach’s conjecture. A simple reasoning is developed to show that any even unsigned integer, belonging to the interval comprised between two successive powers of two, complies with the fact that it can be expressed as the sum of all the odd integers in the interval from 1 up to the nearest upper power of two, and thus as a sum of all pairs of prime numbers within the interval.
TL;DR: In this paper, it was shown that for a quadratic field, there are at most elliptic points on a Shimura curve of Γ 0(p)-type for every sufficiently large prime number p.
Abstract: In a previous article, we proved that for a quadratic field, there are at most elliptic points on a Shimura curve of Γ0(p)-type for every sufficiently large prime number p. This is an analogue of the study of rational points on the modular curve X
0(p) by Mazur and Momose. In this article, we expand the previous result for Shimura curves to the case of number fields of higher degree, which seems unknown for X
0(p).
TL;DR: Wei and Wooley as discussed by the authors improved the estimate for the exponential sum over primes in short intervals when for in the minor arcs, and combined with the Hardy-Littlewood circle method, this enables them to investigate the Waring-Goldbach problem concerning the representation of a positive integer as the sum of th powers of almost equal prime numbers.
Abstract: Let be the von Mangoldt function, be real and . This paper improves the estimate for the exponential sum over primes in short intervals when for in the minor arcs. When combined with the Hardy–Littlewood circle method, this enables us to investigate the Waring–Goldbach problem concerning the representation of a positive integer as the sum of th powers of almost equal prime numbers, and improve the results of Wei and Wooley [On sums of powers of almost equal primes. Proc. Lond. Math. Soc. (3) 111(5) (2015), 1130–1162].
TL;DR: The p-rank of the class group of Q(N^(1/p)) is estimated in terms of the discrete logarithm, with values un F_p, of certain units using the Gross--Koblitz formula and identities on the N-adic Gamma function.
Abstract: Let N and p be two prime numbers > 3 such that p divides N-1. We estimate the p-rank of the class group of Q(N^(1/p)) in terms of the discrete logarithm, with values un F_p, of certain units. Using the Gross--Koblitz formula and identities on the N-adic Gamma function, we explicitly compute these logarithms. A special case (for which we don't have an elementary proof) of our formula is the following: assume there are some integers $a$, $b$ such that N = (a^p+b^p)/(a+b). Then (a+b)*\prod_{k=1}^{(N-1)/2} k^{8k} is a p-th power modulo N. Furthermore we give a new proof which doesn't use modular forms of a result of Calegari and Emerton.
Journal Article•
TL;DR: In this article, it was shown that Thompson's conjecture holds for simple groups L n(2), where 2 n 1 = p is a prime number, and that the set of the conjugacy classes size of L is equal to G.
Abstract: n 1 = p is a prime number. Furthermore, we will show that Thompson's conjecture holds for the simple groups L n(2), where 2 n 1 prime is a prime number. By Thompson's conjecture if L is a nite non-Abelian simple group, G is a nite group with a trivial center, and the set of the conjugacy classes size of L is equal to G , then LG .
TL;DR: In this article, the authors obtained the slopes of the Newton polygons of the L-functions of the exponential sums associated to f ( x ) for any nontrivial finite character χ.
TL;DR: It is shown that for all integers $1
Abstract: Kurepa's conjecture states that there is no odd prime $p$ that divides $!p=0!+1!+\cdots+(p-1)!$. We search for a counterexample to this conjecture for all $p<2^{34}$. We introduce new optimization techniques and perform the computation using graphics processing units. Additionally, we consider the generalized Kurepa's left factorial given by $!^{k}n=(0!)^k +(1!)^k +\cdots+((n-1)!)^{k}$, and show that for all integers $1
TL;DR: In this article, the authors proposed a method to construct integral periodic mask and corresponding scaling step functions that generate non-Haar orthogonal MRA on the local field F ( s ) of positive characteristic p.
Posted Content•
TL;DR: In this paper, the existence and precise definition of higher-order hierarchies, such as division into species, is open to debate among biologists, and a new metric, ''small $s$'' that distinguishes the population number and various data values that are beyond the range of neutral logarithmic populations and are specific to a given species with quantization, by the data from natural environments.
Abstract: The concepts of a population and a species play fundamental roles in biology. The existence and precise definition of higher-order hierarchies, such as division into species, is open to debate among biologists. Here we show a new metric, `small $s$', that distinguishes the population number and various data values that are beyond the range of neutral logarithmic populations and are specific to a given species with quantization, by the data from natural environments. We modify Price equation to introduce the metric. We show prime number could be related with speciation in discontinuity in Riemann zeta function including Bose-Einstein condensation, while prime closed geodesics could represent populations in Selberg zeta function. Calculation of prime closed geodesics $|N(p)|$ leads out non-interacting adaptive species world is in the mode $|N(p)| = 2/3$, while interacting neutral populations is in the mode $|N(p)| = 1$. Combining the calculation with phylogenetical asymmetry, this discrimination gives the investigator whether the observed hierarchy of data represents chaotic populations/non-adaptive species or adaptive species with newly defined critical temperature and Weiss field. The border of fluctuating populations and ordered species is $\Re(s) = 2$, which is proven theoretically, by calculation and by observation. mod 4 of primes corresponding to $\Im(s)$ for zero points of Riemann zeta reveals adaptive/disadaptive situations among the individuals. Furthermore, our model has partially been successful for predicting futures at a unit time after, as transitions of biological phases. The time-dependent fitness function and precise `Hubble parameter' of a fitness space are also predicted by utilizing Schwarz equation.Significance of biological hierarchy is also discussed.In our new model (PzDom model), calculation requires only sets of the numbers of individual densities in time course.
Journal Article•
TL;DR: This paper used three Mersenne prime numbers to construct a new RSA cryptosystem which provides more efficiency and reliability over the network.
Abstract: As the internet evolves and computer networks become accessible, most business transactions are done over the internet. Potential threat to the information is always present and attempts are being made to provide maximum security over the network. One of such new techniques is using multiple prime numbers for RSA cryptosystem, which is not easily breakable. Also, typical prime numbers are used in order to strengthen the algorithm to ensure safe data exchange. In this paper, we used three Mersenne prime numbers to construct a new RSA cryptosystem which provides more efficiency and reliability over the network. Mathematics Subject Classification (2010): 11T71, 14G50, 68P25, 94A60.
Posted Content•
TL;DR: In this article, a path of computable isogenies from an arbitrary simple ordinary abelian surface towards one with maximal endomorphism ring was given, which has immediate consequences for the CM-method in genus 2.
Abstract: Fix a prime number $\ell$. Graphs of isogenies of degree a power of $\ell$ are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called $\mathfrak l$-isogenies, resolving that they are (almost) volcanoes in any dimension. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as $(\ell, \ell)$-isogenies: those whose kernels are maximal isotropic subgroups of the $\ell$-torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.
TL;DR: In this article, it was shown that almost all even integers satisfying certain necessary local conditions are representable as the sum of two primes of the form (x^2+y^2 + 1+1).
Abstract: We study the Goldbach problem for primes represented by the polynomial $x^2+y^2+1$. The set of such primes is sparse in the set of all primes, but the infinitude of such primes was established by Linnik. We prove that almost all even integers $n$ satisfying certain necessary local conditions are representable as the sum of two primes of the form $x^2+y^2+1$. This improves a result of Matomaki, which tells that almost all even $n$ satisfying a local condition are the sum of one prime of the form $x^2+y^2+1$ and one generic prime. We also solve the analogous ternary Goldbach problem, stating that every large odd $n$ is the sum of three primes represented by our polynomial. As a byproduct of the proof, we show that the primes of the form $x^2+y^2+1$ contain infinitely many three term arithmetic progressions, and that the numbers $\alpha p \pmod 1$ with $\alpha$ irrational and $p$ running through primes of the form $x^2+y^2+1$, are distributed rather uniformly.
TL;DR: Several examples of generalized number systems are constructed to compare various conditions occurring in the literature for the prime number theorem in the context of Beurling generalized primes as mentioned in this paper, which is a generalization of the generalized number theorem.
Abstract: Several examples of generalized number systems are constructed to compare various conditions occurring in the literature for the prime number theorem in the context of Beurling generalized primes.
TL;DR: In this article, it was shown that Gal(Qp/Qp) is the etale fundamental group of certain object Z which is defined over an algebraically closed field.
Abstract: Let p be a prime number. In this article we present a theorem, suggested by Peter Scholze, which states that Gal(Qp/Qp) is the etale fundamental group of certain object Z which is defined over an algebraically closed field. As a consequence, p-adic representations of Gal(Qp/Qp) correspond to Qp-local systems on Z. The precise theorem involves perfectoid spaces, [Sch12]. Let C/Qp be complete and algebraically closed. Let D be the open unit disk centered at 1, considered as a rigid space over C, and given the structure of a Zp-module where the composition law is multiplication, and a ∈ Zp acts by x 7→ xa. Let D = lim ←− x 7→xp D.
TL;DR: In this article, the Riemann hypothesis was used to give explicit upper bounds on the difference between consecutive prime numbers, on the assumption of the Rieger hypothesis, and on the expectation that the difference is bounded.
Abstract: In this paper, on the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.
Posted Content•
1 Jan 2016
TL;DR: In this article, it was shown that the invariant field Q(x 1,...,xp) Cp is rational over the cyclic group of order p if and only if the (p 1)-th cyclotomic field has class number one.
Abstract: Let p be a prime number. Let Cp, the cyclic group of order p, permute transitively a set of indeterminates {x1,...,xp}. We prove that the invariant field Q(x1,...,xp) Cp is rational over Q if and only if the (p 1)-th cyclotomic field Q(�p 1) has class number one.
TL;DR: In this paper, it was shown that the Fermat equations do not admit non-trivial solutions for a set of exponents p with Dirichlet density 1/4 and 1/8, respectively.