This is the fourth thread for the Polymath8b project to obtain new bounds for the quantity

$latex \displaystyle H_m := \liminf_{n \rightarrow\infty} (p_{n+m} – p_n),&fg=000000$

either for small values of $latex {m}&fg=000000$ (in particular $latex {m=1,2}&fg=000000$) or asymptotically as $latex {m \rightarrow \infty}&fg=000000$. The previous thread may be found here. The currently best known bounds on $latex {H_m}&fg=000000$ are:

- (Maynard) Assuming the Elliott-Halberstam conjecture, $latex {H_1 \leq 12}&fg=000000$.
- (Polymath8b, tentative) $latex {H_1 \leq 272}&fg=000000$. Assuming Elliott-Halberstam, $latex {H_2 \leq 272}&fg=000000$.
- (Polymath8b, tentative) $latex {H_2 \leq 429{,}822}&fg=000000$. Assuming Elliott-Halberstam, $latex {H_4 \leq 493{,}408}&fg=000000$.
- (Polymath8b, tentative) $latex {H_3 \leq 26{,}682{,}014}&fg=000000$. (Presumably a comparable bound also holds for $latex {H_6}&fg=000000$ on Elliott-Halberstam, but this has not been computed.)
- (Polymath8b) $latex {H_m \leq \exp( 3.817 m )}&fg=000000$ for sufficiently large $latex {m}&fg=000000$. Assuming Elliott-Halberstam, $latex {H_m \ll m e^{2m}}&fg=000000$ for sufficiently large $latex {m}&fg=000000$.

While the $latex {H_1}&fg=000000$ bound on the Elliott-Halberstam conjecture…

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