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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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.
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.