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Legendre's conjecture, asserting that there is at least one prime number between the squares of any two consecutive integers, seems likely to be true, but remains unproved. Wikipedia states that there are no counterexamples up to $4\times 10^{18}$.

My question goes to what degree the conjecture must be weakened in order to provide a provable statement: What is the minimum value of $\alpha\in \mathbb R$ such that the interval $[n^2,(n+\alpha)^2]$ can be proved to contain a prime number? Presumably, $\alpha \ge 1$.

An answer to this question suggests without citation or demonstration that the weakened conjecture holds for $\alpha=2$. I have not located any literature that addresses this question, so I would appreciate learning of references if any exist.

Keith Backman
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    I don't see anything in the linked question which suggests that the weaker conjecture has been proved. If you are speaking of the answer posted by the user "Charles", then note that all that is claimed is that the weaker conjectures can't be proven (yet) even if you assume the GRH. In any case, I'd be surprised if any such statement were provable with current techniques. – lulu Mar 29 '20 at 20:24
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    I agree with @lulu: Also note the section on partial results in the Wikipedia article on Legendre's conjecture. I'd expect it to mention any result of the form you're looking for. – joriki Mar 29 '20 at 21:10
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    On closer reading, I see that @lulu's take on Charles answer is correct, so even less was known than I imagined when I asked my question. – Keith Backman Mar 29 '20 at 21:22
  • The best known upper bound for a prime gap after $p_n$ is $g_n\le p_n^{0.525}$ for sufficiently large $p_n$ although it is conjectured that $g_n<\ln^2(p_n)$ holds for $p_n>7$ – Peter Mar 30 '20 at 13:22

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