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Note for the Prime Gaps
Revision notes
I think Lemma 3 is flawed in the previous version. However, I was able to overcome that problem and demonstrate a partial result. In fact, in this new version we show that the Andrica conjecture should be true for every sufficiently large gap between primes. We changed the abstract, keywords, content and references. This current version remains a good contribution to the study of the distribution of prime numbers in their gaps.
Abstract
A prime gap is the difference between two successive prime numbers. The nth prime gap, denoted $g_{n}$ is the difference between the (n + 1)st and the nth prime numbers, i.e. $g_{n}=p_{n+1}-p_{n}$. There isn't a verified solution to Andrica's conjecture yet. The conjecture itself deals with the difference between the square roots of consecutive prime numbers. While mathematicians have showed it true for a vast number of primes, a general solution remains elusive. We consider the inequality $\frac{\theta(p_{n+1})}{\theta(p_{n})} \geq \sqrt {\frac{p_{n+1}}{p_{n}}}$ for two successive prime numbers $p_{n}$ and $p_{n+1}$, where $\theta(x)$ is the Chebyshev function. In this note, under the assumption that the inequality $\frac{\theta(p_{n+1})}{\theta(p_{n})} \geq \sqrt {\frac{p_{n+1}}{p_{n}}}$ holds for all $n \geq 1.3002 \cdot 10^{16}$, we prove that the Andrica's conjecture is true. Since $\frac{\theta(p_{n+1})}{\theta(p_{n})} \geq \sqrt {\frac{p_{n+1}}{p_{n}}}$ holds indeed for large enough prime number $p_{n}$, then we show that the statement of the Andrica's conjecture can always be true for all primes greater than some threshold.
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References
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