---

Statement

Theorem 1.3 (p. 3).

$$\liminf_{n}\,(p_{n+1}-p_n)\le600 .$$

That is, $p_{n+1}-p_n\le600$ for infinitely many $n$. The paper stresses (p. 3) that the proof uses none of Zhang's technology and relies only on the Bombieri--Vinogradov theorem, and that the constant 600 is not optimal.

Source. J. Maynard, Small gaps between primes, Ann. of Math. (2) 181 (2015), no. 1, 383--413, doi:10.4007/annals.2015.181.1.7, read in the arXiv:1311.4600v3 preprint (28 October 2019) identified on the source card; the pages cited are the preprint's printed pages, not the journal's. Theorem 1.3 on p. 3, the proof on p. 6.

Read depth. Claims checked: the statement and the deduction on p. 6 were read clause by clause. The numerical bound $M_{105}>4$ (Proposition 4.3 (2), proved in Section 8, pp. 21--24) was read for its structure and not recomputed, and the admissible 105-set of diameter 600 listed in the footnote on p. 6 was not checked. Nothing here is independently reviewed.

Proof pointer

P. 6. Take $k=105$. Proposition 4.3 (2) gives $M_{105}>4$, and Bombieri--Vinogradov gives level of distribution $\theta=1/2-\epsilon$, so $\theta M_{105}/2>1$ for small $\epsilon$. Then Proposition 4.2 gives $\liminf(p_{n+1}-p_n)\le h_{105}-h_1$ for any admissible $\{h_1<\dots<h_{105}\}$, and an admissible set with $h_{105}-h_1=600$, credited to unpublished computations of Thomas Engelsma, is listed in the footnote.

Dependencies

Proposition 4.2 and Proposition 4.3 (2) of the same paper; the Bombieri--Vinogradov theorem.