Bounded prime gap constant

Upper bound, last-listed row

| $240$ | [Sta26] | Stadlmann: independent proof using the Bombieri–Vinogradov theorem combined with newer equidistribution estimates for smooth moduli. Preprint of 31 August 2026. |value changed 2026-09-26

Lower bound, last-listed row

| $2$ | Trivial | $p\_{n+1} - p\_n \ge 2$ for every $n \ge 2$. Equality, $H\_1 = 2$, is the twin prime conjecture. |tracked since 2026-09-02 — no movement seen yet

Checked against material for its cited source

• Another row, not one shown above: bound row $4680$ line 20 citing [Pol14a] — the claim we checked: that the cited Polymath8a paper gives the upper bound this row lists, 4680, by optimizing Zhang's equidistribution estimates, as the row's comment says. What we read could not settle it for the edition cited: the journal edition says the bound, lim inf of p(n+1) − p(n) at most 4680, was established by the paper's first version, and that the present version contains only the equidistribution estimates. The earlier version, arXiv 1402.0811v2, titled with "and bounded gaps between primes", states 4680 as its Theorem 1.3. source link (read in the journal edition the entry cites, Algebra & Number Theory 8 (2014): the introduction and the opening of Section 2 (pp. 2067–2075) and the references, from the publisher's PDF; and in the paper's earlier version, arXiv preprint version 2 (12 Jul 2014), pp. 1–6 and 10–20). Selected: drawn by its position in the list of cited rows, not because it looked wrong.
• Another row, not one shown above: bound row $70\,000\,000$ line 19 citing [Zha14] — the claim we checked: that the cited paper proves the bound 70,000,000 by the equidistribution estimate for smooth moduli beyond the Bombieri–Vinogradov range that the row's comment names, and is the first proof of finiteness, as the comment says. What we read supports the bound and the method. That it is the first proof of finiteness is the paper's own account of earlier work, not checked independently. source link (read in the body of the published Annals of Mathematics paper (179 (2014), 1121–1174), the publisher's own PDF: the abstract on p. 1121, Theorem 1 on p. 1122 and the method sketch on pp. 1125–1126, pp. 1121–1122 read from page images). Selected: drawn by its position in the list of cited rows, not because it looked wrong.

2 bound row(s) here have been checked against material for their cited source (2 drawn by position, 0 chosen because something already looked wrong). Across this whole ledger 70 row(s) have been drawn by position, against 691 rows that name a source, counted on 2026-09-27: 64 were checked against material for their cited sources and for 6 the cited source could not be read at all. 5 more were chosen for suspicion and checked; they are counted apart, because a set selected for suspicion carries no rate. A row not listed here has NOT been checked.