Largest known integer $m$ such that Legendre's conjecture is true up to $m$

Question

The Legendre's conjecture states that the inteval $(m^2,(m+1)^2)$ contains a prime $p_n$ for each $m$. I wante to find the largest known integer $m$ such that this conjecture is true up to $m$. Wikipedia (https://en.wikipedia.org/wiki/Legendre%27s_conjecture#cite_ref-3) assert that this number is $4×10^{18}$ but I see that this is related to the Goldbach conjecture . I very confused about these facts.

I suspect you want the largest $m$ for which it is known that this conjecture holds up to $m$. — Wojowu, Nov 21 '15 at 08:16

score 2 · Accepted Answer · answered Nov 21 '15 at 08:26

Let $P$ be the largest known prime and $m=\lfloor\sqrt P\rfloor$. Then $$ m^2<P<(m+1)^2. $$ Since $P$ is the largest known prime, $m$ is the largest integer known for which Legendre's conjecture holds.

A different question would be to ask for the largest integer $m$ such that the conjecture holds for all smaller integers.

Largest known integer $m$ such that Legendre's conjecture is true up to $m$

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