# The Difference Between Consecutive Primes, II

@article{Baker2001TheDB,
title={The Difference Between Consecutive Primes, II},
author={Roger C. Baker and Glyn Harman and Janos Pintz},
journal={Proceedings of the London Mathematical Society},
year={2001},
volume={83}
}
• Published 1 November 2001
• Mathematics
• Proceedings of the London Mathematical Society
The authors sharpen a result of Baker and Harman (1995), showing that [x, x + x0.525] contains prime numbers for large x. An important step in the proof is the application of a theorem of Watt (1995) on a mean value containing the fourth power of the zeta function. 2000 Mathematical Subject Classification: 11N05.
421 Citations

### Short Effective Intervals Containing Primes and a property of the Riemann Zeta Function ζ

In this paper, we prove the existence of primes in the interval ]�, � + 2 �] by inducing an inequality which defines the lower bound of number of primes in the interval ]�, � + 2 �] and suggest an

### Arithmetic Properties of Blocks of Consecutive Integers

• Mathematics
• 2016
This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of

### Gaussian primes in Narrow sectors

• Mathematics
• 2001
The purpose of this paper is to show how a sieve method which has had many applications to problems involving rational primes can be modified to derive new results on Gaussian primes (or, more

### On Proving of Diophantine Inequalities with Prime Numbers by Evaluations of the Difference between Consecutive Primes

Using as the working hypothesis of an evaluation of the difference between primes $p_{n+1} - p_n = O(\sqrt{p_n})$ we represent in detail the proofs of Legendre's and Oppermann's conjectures.

### REPRESENTING AN INTEGER AS THE SUM OF A PRIME AND THE PRODUCT OF TWO SMALL FACTORS

• Mathematics
• 2020
Let c > 0.55. Every large n can be written in the form p +ab, where p is prime, a and b are significantly smaller than x^1/2 and ab is less than n^c. This strengthens a result of Heath-Brown, which

### Primes between consecutive squares

Abstract. A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of

### Squarefree Integers And Extreme Values Of Some Arithmetic Functions

A study of the Dedekind psi function concludes that its extreme values are supported on the subset of primorial integers N_k = 2*3***p_k, where p_k is the kth prime. In particular, the inequality

### A Structure Theorem for Positive Density Sets Having the Minimal Number of 3-term Arithmetic Progressions

Assuming the well-known conjecture that [x,x+x^t] contains a prime for t > 0 and x sufficiently large, we prove: For 0 0 as s -> 0. A curious feature of the proof is that Behrend's construction on

### Diophantine Inequalities as a Problem of Difference between Consecutive Primes

In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach

### Goldbach Numbers in Short Intervals -- A Nonnegative Model Approach

We decrease the length of the shortest interval for which almost all even integers in it are the sum of two primes. This is achieved by applying a version of the Circle Method that uses two minorants