Large gaps between consecutive prime numbers (1408.4505v2)

Abstract: Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.

Large Gaps Between Consecutive Prime Numbers

The paper "Large Gaps Between Consecutive Prime Numbers," authored by Kevin Ford, Ben Green, Sergei Konyagin, and Terence Tao, investigates the maximal gaps between prime numbers and provides new results that contribute to the understanding of the distribution of primes. This paper addresses a question posed by Erdős and demonstrates that the size of the largest gap between consecutive primes below a number $X$ can exceed $$f(X) \log X \log \log X \log \log \log X / (\log \log \log X)$$, where $f(X)$ is a function that tends to infinity with $X$.

Key Findings and Numerical Results

One of the significant results in this paper is Theorem 1, which asserts that for any sufficiently large $X$, there exist at least $R \log X / (\log X)$ consecutive composite numbers not exceeding $X$. This implies that the gaps between consecutive primes can be considerably large. The paper achieves this result by leveraging various advanced mathematical tools, including probabilistic models, arithmetic progressions, and sieve-theoretic techniques.

The research surpasses previous bounds set by notable mathematicians like Erdős and Rankin by demonstrating that the gaps between consecutive primes can indeed be much larger than earlier anticipated. The paper also explores unconditional upper bounds, reporting that the best known bound, $G(X) \ll X^{0.525}$, is far from conjectured values.

Implications and Future Directions

The implications of discovering larger gaps between consecutive primes are both theoretical and practical in nature. On the theoretical side, these results contribute to the understanding of prime distribution and challenge existing probabilistic models such as Cramér’s conjecture. The work raises questions about the nature of prime gaps, potentially guiding future research towards better models or conjectures.

Practically, the findings could influence computational number theory, particularly in areas such as cryptography, where the understanding of prime distribution is crucial. The paper sets the stage for developing new techniques to compute or estimate large gaps between prime numbers efficiently, an area that could benefit both theoretical investigations and practical applications.

Techniques and Methodology

The authors employ sophisticated arguments combining probabilistic models with analyses of arithmetic progressions composed entirely of primes. They also adopt sieve-theoretic techniques to explore linear equations in primes with large shifts. The paper introduces a method to randomly refine subsets of primes, ensuring that certain arithmetic conditions are met—an approach that could serve as a foundation for further developments in sieve theory and prime gap assessments.

Conclusion

The research on large gaps between consecutive prime numbers presented in this paper is a substantial step forward in mathematical number theory. With its rigorous methodology and significant findings, it poses new challenges and opportunities for researchers interested in prime distributions. The implications of this paper are broad and could influence various directions in future research, potentially leading to a deeper understanding of primes and their complex behavior in the context of number theory and computational applications.