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Is this problem still open? I know that Henri Brocard conjectured that there are at least four primes in the interval between each pair of consecutive squares of primes from nine onward. http://mathworld.wolfram.com/BrocardsConjecture.html

I know that Brocard's conjecture remains open. I am asking if there is yet a proof for only one prime between each pair of consecutive squares of primes. Is this a problem with or without a name? Where can I find a reference about it or any information on progress made on it?

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    $\begingroup$ It sounds like Legendre's conjecture restricted to the primes. $\endgroup$ Apr 5 '16 at 6:50
  • $\begingroup$ Right. It is similar to Legendre's conjecture, but weaker. $\endgroup$ Apr 5 '16 at 6:52
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A reference, among others, on such problems is the paper On Legendre’s, Brocard’s, Andrica’s, and Oppermann’s Conjectures by German Andres Paz. His "Conjecture $1$" related to Brocard's conjecture as follows. Let denote $p_n$ and $p_{n+1}$ two successive primes greater than $2$. Since $p_{n+1}-p_n\ge 2$, we know that there is a positive integer $k$ with $p_n<k<p_{n+1}$. Hence we have $p_n^2<k^2<p_{n+1}^2$. By conjecture $1$ there are at least two primes between $p_n^2$ and $k^2$, and between $k^2$ and $p_{n+1}^2$. Hence we have at least $4$ primes between $p_n^2$ and $p_{n+1}^2$. Hence Brocard's conjecture follows.

So I think, Conjecture $1$, Brocard's conjecture, and "your conjecture" (which is a variant of these) are still open.

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