# State of the Art — Axis #10: Prime Deserts (large prime gaps) ## What is known (checked against the literature, 2026-08-06) | Quantity | Known record | Source | |---|---|---| | Largest **maximal** prime gap | 1854, after p = 101412319996363309069 (85th maximal gap, Robert Smith / Brian Kehrig code, 2026) | primerecords.dk, t5k.org GapsTable | | Exhaustive verification bound for maximal gaps | ≈ 2×10^19 (beyond 2^64) | t5k.org/notes/GapsTable.html | | Largest known gap (any size, PRP endpoints) | 16,045,848 near a 385,713-digit number (A. Höglund, 2024) | primerecords.dk/primegaps/megagap3.htm | | Highest known **merit** g/ln(p) | 41.93878… (gap 8350 after an 87-digit prime, Gapcoin network, 2017) | primegap-list-project | | Highest Cramér–Shanks–Granville ratio g/ln²(p) | 0.9206… (Nyman's gap 1132 after 1693182318746371, 1999) | MathWorld Prime Gaps | | Theory (lower bound) | Ford–Green–Konyagin–Maynard–Tao 2016: max gap ≥ c·(ln x · lnln x · lnlnlnln x)/lnlnln x | published | | Theory (upper bound) | Baker–Harman–Pintz: gap = O(x^0.525); RH gives O(√x ln x) | published | | Heuristic | Cramér: limsup g/ln²p = 1; Granville correction: ≥ 2e^{-γ} ≈ 1.1229 | published | | Jumping champion | 6 is the most common gap from ≈ 947 up to ≈ 1.7×10^35 (conjectured, verified in ranges); then 30 | published (Odlyzko–Rubinstein–Wolf) | ## What would count as NEW here 1. **A new maximal gap** — requires exhaustive sieving beyond 2×10^19: **out of reach** locally; not the goal. 2. **A gap with merit > 41.94** — requires massive targeted search (Gapcoin-scale): out of reach in one cycle; a merit > 20 gap found by our own sieve pipeline would still only be "personal best" territory (thousands of merit-20+ gaps are catalogued). Any gap we find must be checked against the primegap-list-project tables before any claim. 3. **Structural conjectures about gap statistics** (residues of maximal-gap primes, corrections to Cramér in arithmetic progressions, gap-pair correlations) — realistic: findings are likely "(a) probably known", but a precise, massively tested statement with clean data is a legitimate output. 4. **Certified desert constructions** (covering systems): any construction is reproducible and certifiable, but primorial/covering constructions have low merit (~2–4) — value is the **certificate pipeline**, not records. ## Exact thresholds to beat (for honesty in any claim) - Maximal gap: must exceed 1854 AND be proven maximal (impossible without exhaustive sieve > 2×10^19) — or simply reproduce/verify the known table below our compute bound (verification, not record). - Merit record: > 41.93878. - CSG ratio record: > 0.92064. - First-occurrence gaps: the Nicely / primegap-list-project tables catalogue first known occurrences for every even gap; before claiming a "first occurrence", check those tables. Sources: [t5k.org GapsTable](https://t5k.org/notes/GapsTable.html), [primerecords.dk top-20](http://primerecords.dk/primegaps/gaps20.htm), [primerecords.dk megagap](http://primerecords.dk/primegaps/megagap3.htm), [primegap-list-project merit record](https://primegap-list-project.github.io/record/2017/12/31/new-prime-gap-of-maximum-merit/), [MathWorld Prime Gaps](https://mathworld.wolfram.com/PrimeGaps.html), [Wikipedia Prime gap](https://en.wikipedia.org/wiki/Prime_gap), [Nicely first-occurrence tables](https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html)