In the present work we prove a common generalization of Maynard–Tao’s recent result about consecutive bounded gaps between primes and of the Erdős–Rankin bound about large gaps between consecutive primes. The work answers in a strong form a 60-year-old problem of Erdős, which asked whether the ratio of two consecutive primegaps can be infinitely often arbitrarily small, and arbitrarily large, respectively. This is proved in the paper in a stronger form that not only \(d_{n} = p_{n+1} - p_{n}\) can be arbitrarily large compared to dn+1 but this remains true if dn+1 is replaced by the maximum of the k differences \(d_{n+1},\ldots,d_{n+k}\) for arbitrary fix k. The ratio can reach c(k) times the size of the classical Erdős–Rankin function with a constant c(k) depending only on k.

On the Ratio of Consecutive Gaps Between Primes

In the present work we discuss several problems concerning the difference of primes, primarily regarding the difference of consecutive primes. Most of them were either initiated by Paul Erdős (sometimes with coauthors), or were raised ahead of Erdős; nevertheless he was among those who reached very important results in them (like the problem of the large and small gaps between consecutive primes).

Paul Erdős and the Difference of Primes

Consider the comparison of two semiprimes phpk and pipj, where pn is the nth prime. If h pipj is possible, and the actual direction be...

Random Ordering of Semiprimes

A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size. The method is formulated using the linear multistep method, but the new method possesses the desirable feature of the one-step method. The implementation is based on the predictor and corrector formulas in the mode. The stability and precision of this method will also be analyzed and deliberated. Numerical results are given to show the efficiency of the proposed method and will be compared with the existing method.

/pdf/direct-two-point-block-one-step-method-for-solving-general-hyitf4wsav.pdf

Direct Two-Point Block One-Step Method for Solving General Second-Order Ordinary Differential Equations

Publisher Summary This chapter discusses the problems and results on combinatorial number theory. It discusses number theoretic problems that are of combinatorial nature. Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers along with the properties of objects made out of integers. Integers can be considered either in themselves or as solutions to equations. Questions in number theory are often understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. Real numbers can also be studied in relation to rational numbers.

Problems and results on combinatorial number theory III

Some Problems On Consecutive Prime Numbers

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Some Problems On Consecutive Prime Numbers