Consecutive Prime Numbers Research Articles

Un système dynamique lié à la suite des nombres premiers

We study the dynamical system defined by the map Φ: ]0,1]→ ]0,1] , where Φ( x)= px−1 on ]1/ p,1/ q] if q and p are consecutive prime numbers. We relate the existence of an absolutely continuous invariant measure to ergodicity of a Markov chain P on the union of orbits stemming from numbers 1/ p ( p prime). We prove that ergodicity of P implies ergodicity of Φ. We establish a link between transfer probabilities of order n and some sets of sequences of the symbolic dynamic. This leads to a way of computing these coefficients using Monte Carlo method. We propose an algorithm which leads to a density indicating a good experimental fit with a typical orbit.

Hall normalization constants for the Bures volumes of then-state quantum systems

We report the results of certain integrations of quantum-theoretic interest, relying, in this regard, upon recently developed parameterizations of Boya et al of the n x n density matrices, in terms of squared components of the unit (n-1)-sphere and the n x n unitary matrices. Firstly, we express the normalized volume elements of the Bures (minimal monotone) metric for n = 2 and 3, obtaining thereby "Bures prior probability distributions" over the two- and three-state systems. Then, as an essential first step in extending these results to n > 3, we determine that the "Hall normalization constant" (C_{n}) for the marginal Bures prior probability distribution over the (n-1)-dimensional simplex of the n eigenvalues of the n x n density matrices is, for n = 4, equal to 71680/pi^2. Since we also find that C_{3} = 35/pi, it follows that C_{4} is simply equal to 2^{11} C_{3}/pi. (C_{2} itself is known to equal 2/pi.) The constant C_{5} is also found. It too is associated with a remarkably simple decompositon, involving the product of the eight consecutive prime numbers from 2 to 23. We also preliminarily investigate several cases, n > 5, with the use of quasi-Monte Carlo integration. We hope that the various analyses reported will prove useful in deriving a general formula (which evidence suggests will involve the Bernoulli numbers) for the Hall normalization constant for arbitrary n. This would have diverse applications, including quantum inference and universal quantum coding.

On smooth gaps between consecutive prime numbers

Investigations concerning the gaps between consecutive prime numbers have long occupied an important position on the interface between additive and multiplicative number theory. Perhaps the most famous problem concerning these gaps, the Twin Prime Conjecture, asserts that the aforementioned gaps are infinitely often as small as 2. Although a proof of this conjecture seems presently far beyond our reach (but see [5] and [10] for related results), weak evidence in its favour comes from studying unusually short gaps between prime numbers. Thus, while it follows from the Prime Number Theorem that the average gap between consecutive primes of size about x is around log x , it is now known that such gaps can be infinitely often smaller than 0–249 log x (this is a celebrated result of Maier [12], building on earlier work of a number of authors; see in particular [7], [13], [3] and [11]). A conjecture weaker than the Twin Prime Conjecture asserts that there are infinitely many gaps between prime numbers which are powers of 2, but unfortunately this conjecture also seems well beyond our grasp. Extending this line of thought, Kent D. Boklan has posed the problem of establishing that the gaps between prime numbers infinitely often have only small prime divisors, and here the latter divisors should be small relative to the size of the small gaps established by Maier [12]. In this paper we show that the gaps between consecutive prime numbers infinitely often have only small prime divisors, thereby solving Boklan's problem. It transpires that the methods which we develop to treat Boklan's problem are capable also of detecting multiplicative properties of more general type in the differences between consecutive primes, and this theme we also explore herein.

A Note on Consecutive Prime Numbers

A Note on Consecutive Prime Numbers

On the distance between consecutive prime ideal numbers in sectors

On the distance between consecutive prime ideal numbers in sectors

On the difference between consecutive prime numbers

On the difference between consecutive prime numbers

Some problems on consecutive prime numbers

Some problems on consecutive prime numbers

Convergence of successive substitution starting procedures

A. Schinzel and W. Sierpifnski [1] conjectured that there exist arbitrary long arithmetic progressions formed of consecutive prime numbers. Sierpi'ski stated in [2] that a progression of five consecutive primes had not yet been found. A direct computer search showed that the first such progression has the common difference d = 30 and begins with the prime 9,843,019. The first progression of six consecutive primes begins with 121,174,811 and also has d = 30. Up to the limit 3 X 108 there are 25 other progressions of five consecutive primes, all with d = 30; there are no other progressions of six consecutive primes. The referee points out that recently a much larger quintuplet, beginning with 10000024493, and again having d = 30, was recorded [3], but without reference to Sierpifnski's remark. The smaller set that we found, and the single sextuplet, may still be worth recording.

The Difference between Consecutive Prime Numbers V

Let pn denote the nth prime and let ε be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n,where, for k≧1, logk+1x = log (logk x) and log1x = log x. In a recent paper (4) Schönhage has shown that the constant ⅓ may be replaced by the larger number ½eγ, where γ is Euler's constant; this is achieved by means of a more efficient selection of the prime moduli used. Schönhage uses an estimate of mine for the number B1 of positive integers n≦u that consist entirely of prime factors p≦y, whereHerer x is large and α and δ are positive constants to be chosen suitably.

The Difference Between Consecutive Prime Numbers. IV

The Difference Between Consecutive Prime Numbers. IV

The difference between consecutive prime numbers. IV

The difference between consecutive prime numbers. IV

The Difference between Consecutive Prime Numbers, III

Journal of the London Mathematical SocietyVolume s1-22, Issue 3 p. 226-230 Notes and papers The Difference between Consecutive Prime Numbers, III R. A. Rankin, R. A. Rankin Clare College CambridgeSearch for more papers by this author R. A. Rankin, R. A. Rankin Clare College CambridgeSearch for more papers by this author First published: July 1947 https://doi.org/10.1112/jlms/s1-22.3.226Citations: 6AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes1-22, Issue3July 1947Pages 226-230 RelatedInformation

The difference between consecutive prime numbers. II

In a previous paper, under the same title, I considered the problem of how far apart two consecutive primes can be. The present paper is concerned with the opposite question. How near together can large primes lie? The published literature on this subject is scanty and, though interesting, is mainly negative in character. It appears to be very difficult to give any answer that is not trivial, or that is at all illuminating.

The Difference between Consecutive Prime Numbers

The Difference between Consecutive Prime Numbers

On the order of magnitude of the difference between consecutive prime numbers

On the order of magnitude of the difference between consecutive prime numbers