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I checked some differences between square roots of various natural numbers and I am wondering what is required to prove Andrica's conjecture. Would a tighter upper bound for the prime gap above $n$ be sufficient? Would prime gaps have to be bounded by the product of a constant and the square root of $n$ for this conjecture to be proved?

Does Andrica's conjecture imply anything about prime gaps?

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    $\begingroup$ It is a prime gap conjecture. $\endgroup$ – André Nicolas Oct 7 '14 at 5:00
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    $\begingroup$ Both questions are answered in the preface to the Wikipedia article. $\endgroup$ – Slade Oct 7 '14 at 5:05
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    $\begingroup$ So my recommendation, Jeffrey, would be to read the Wikipedia piece and then post a summary as an answer. $\endgroup$ – Gerry Myerson Oct 7 '14 at 6:20
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    $\begingroup$ Ok. I had read through some webpages about Andrica's conjecture but was I was not clear about what is required to prove it. Now that I know the answers are there, I will seek them out. $\endgroup$ – Jeffrey Young Oct 7 '14 at 12:36
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Andrica's conjecture is $$ g_n<2\sqrt{p_n}+1, $$ or equivalently $$ \sqrt{p_n+g_n}-\sqrt{p_n}<1, $$ where $g_n=p_{n+1}-p_n$ is the n-th prime gap.

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Yes, Andrica's conjecture can be proven by proving a tight enough bound for the prime gap above $n$.

If prime gaps above $n$ for all $n \geq 7$ are bounded by the constant 2 multiplied by a prime < $\sqrt n$, then Andrica's conjecture is true because for Andrica's conjecture to be proven, it would have to be proven that the prime gap above $n$ is bounded by the constant 2 multiplied by $\sqrt n$.

As to what Andrica's conjecture implies will require investigation.

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  • $\begingroup$ It certainly does not imply that prime gaps are bounded! $O(\sqrt n)$ is a long way from $O(1).$ $\endgroup$ – Charles Oct 7 '14 at 17:39

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