cert-machine · Erdős #852 · the constant, finally against the data

A conjecture nobody had checked against the numbers

Erdős #852 asks how long a run of consecutive prime gaps can stay pairwise distinct. Its discussion thread conjectures h(x) ~ c₀·log x. This repository certified c₀ to 61 digits last week. The exact record data has been in the OEIS since 2002. Nobody had put the two in the same room — so we did, and the answer depends on a reading of the problem statement that turns out to be decidable from the data.

tl;dr

The finding. Read as the problem is written — "for some n < x", where n indexes a prime — the certified c₀ lands inside 12 of 22 plateau bands of the exact record data, with no drift across 8 decades. Read the other way, with x as a bound on the prime itself, the same data gives a ratio of about 1.1450 and falling away from c₀. The data picks the reading. It also bears on Erdős's own question: he asked whether h(x) = o(log x), and the ratio here is flat near c₀, not decaying.
The mechanism. Integers only — no floats anywhere in the computation. A run is a window of gaps with no repeat; a record is the first index whose window beats every earlier one, found by a streaming two-pointer scan. Because h is a step function whose records ARE its jumps, sampling the ratio at records reads only the tops of the steps and flatters it, so every plateau is reported as a band — the ratio at the step's start and at its end — and c₀ is asked to lie inside, not to match a corner.
Check it. node instruments/erdos852h/h.js 3e8 — the deep run in certs/erdos852-h-records.json, 4042.2 s to 5e+11. Our first 27 record indices and start primes are checked term for term against the sha-pinned bytes of A078515 and A079889; a disagreement refuses this page.

records recomputed here

independent of OEIS — a streaming integer scan, written from the problem statement

agreement with OEIS

27 / 27

A078515 indices and A079889 start primes, term for term, against sha-pinned bytes

longest run found

a run of 31 consecutive prime gaps, pairwise distinct — an exhibit anyone can check

c₀ inside the band

12 of 22

the honest test: does the certified constant lie between each plateau's high and low ratio

the other reading

≈ 1.1450

x as a bound on the prime rather than on n — sits well below c₀ and is not closing

beyond the published terms

record runs longer than anything in A078515 / A079007 / A079889, each re-proved here from nothing

proved

nothing

#852 is explicitly not resolvable by a finite computation. This is evidence, and the page says so throughout

§1 · the two halves

A constant with no data, and data with no constant

half	where it lived	what it says
the constant	the #852 discussion thread, published 2026-04-24	h(x) ~ c₀·log x, with c₀ the unique positive root of I₀(c) = 1. Certified here to 61 digits: c₀ = 1.32322827686394946902…
the data	OEIS A053597 / A078515 / A079007 / A079889, since 2002	the exact record runs — every index at which a longer pairwise-distinct run of gaps begins, and the prime that opens it
the crossing	nowhere	the problem page lists the sequences in its OEIS box and the thread states the constant; neither does anything with the other

That gap is the whole opportunity, and it needed no new mathematics to close — only the willingness to compute one side and look at the other. We recomputed the records from scratch rather than reading them out of the OEIS, so the comparison has two independent witnesses to the same integers: a sequence from 2002 and a scan written this week from the problem statement. They agree on all 27 terms we can both reach.

§2 · the reading

The problem says "for some n < x", and n indexes a prime

As published, #852 reads: "Let dₙ = pₙ₊₁ − pₙ … let h(x) be maximal such that for some n < x the numbers dₙ, …, dₙ₊ₕ₍ₓ₎₋₁ are all distinct." The bound is on n, and n is an index into the primes — so x counts primes rather than measuring size. It is an easy thing to read past, and the conjecture means something different under each reading. The data is not ambiguous about which one it belongs to.

The same exact data under the two readings. c₀ is a dashed reference because it is a PREDICTION, not a measurement — predicted and decided never share a typography on these pages.

x	h(x), index reading	ratio	h(x), prime reading	ratio	c₀·log x
1e+4	13	1.4115	9	0.9772	12.19
1e+5	15	1.3029	13	1.1292	15.23
1e+6	18	1.3029	15	1.0857	18.28
1e+7	21	1.3029	17	1.0547	21.33
1e+8	25	1.3572	21	1.1400	24.37
1e+9	26	1.2546	22	1.0616	27.42
1e+10	30	1.3029	26	1.1292	30.47
1e+11	31	1.2239	29	1.1450	33.52

§3 · the honest test

h is a step function, so the ratio is a band — not a point

Between one record closing and the next, h(x) is constant while log x keeps growing, so the ratio falls steadily from a high at the step's start to a low at its end. Quoting the ratio at the records — which is where it is highest — would be picking the flattering corner of every step. So each plateau is drawn as the full band it occupies, and the test is whether c₀ lies inside.

Each row is one plateau of h. The vertical mark is the certified c₀. It lands inside 12 of 22 bands; where it misses, it misses narrowly and in both directions — 7 bands sit entirely below it and 3 entirely above.

§4 · the extension

Past the last published term

The record data stopped where other people’s computations stopped: A079007 ends at a run of 30 gaps opening at 196948778371, and A078515 and A079889 end at the same place. Our scan reproduced that term exactly and kept going.

run length	opening prime	index n	status
30	196948778371	7889803997	the last term the literature holds — re-proved here as a control
31	263552821783	10435962861	new beyond every published term

The exhibit, re-proved at this build by a program sharing no line with the scan that found it. 263552821783 is prime, and the 31 consecutive gaps that follow it are pairwise distinct:

76, 8, 34, 36, 26, 106, 38, 82,
18, 54, 50, 16, 62, 70, 14, 48,
42, 64, 6, 104, 24, 12, 66, 22,
30, 96, 2, 10, 74, 4, 32

The run spans the primes 263552821783 to 263552823109. It is exactly 31 long and not longer, because the next gap is 4, which already appears in the list. Anyone can check every word of that in a minute, which is the point of an existence claim.

two claims, and they are not equally strong

That h reaches 31 at this prime is re-proved above by an independent implementation — BigInt Miller–Rabin and a set — and does not depend on the scan at all. That this is the smallest index achieving it is a statement about every index below it, and only the exhaustive scan can speak to that. The scan reproduced all 27 published terms exactly, which is the best evidence available for it, and it is still evidence rather than an independent proof. Both are stated separately here for that reason.

what this page does NOT claim

We did not prove the asymptotic and we could not have: the #852 page states plainly that the problem "cannot be resolved with a finite computation", and that is correct. What is here is a finite, exact computation placed against a published conjecture and a certified constant, reported with its sampling bias removed. It is consistent with h(x) ~ c₀·log x under the index reading and inconsistent with it under the prime reading, over the range we can reach. Consistency over 8 decades is evidence, not a theorem, and the ratio's misses — 7 bands below c₀ and 3 above — are shown rather than smoothed. The one hard result on this page is the agreement: 27 record indices and 27 start primes, recomputed from scratch, match sequences published in 2002 exactly.

§5 · re-run it

Integers, streaming, no dependencies

git clone https://github.com/carlostoledo1891/cert-machine
cd cert-machine
node instruments/erdos852h/h.js 3e8

The scan streams: a segmented sieve feeds a two-pointer window, and only a 1024-entry last-seen table is held, so memory does not grow with the limit. The certified constant it is compared against lives in certs/erdos852-certificate.json and is re-checked by python3 verify/verify_erdos852.py — the subject of the companion page on the constant itself, where one of the two published decimals was refuted at its 12th digit.