Erdoes #141 - working report - Erdős Problem a Day

Erdős problem 141: live-page audit, exact census, and a conditional reduction

Date of audit: 2026-07-26 (UTC)

Outcome

[d] I did not find an unconditional solution of the problem or a new

CPAP-11. I did obtain an exact census through \(10^9\): the longest block in

that range has length \(6\), it is unique, and no block of \(7\) consecutive

primes in arithmetic progression has last term at most \(10^9\).

[b] I independently certified the published CPAP-10: the companion

standard-library-only Python program contains ten complete ECPP certificates,

verifies every elliptic-curve operation, terminates each certificate in a

deterministic 64-bit primality proof, and proves every one of the other 1881

integers in the enclosing interval composite. This is a new self-contained

verification artifact for a known construction, not a new CPAP-10.

[a,b] I also give a corrected CRT construction proving that Dickson's

conjecture (equivalently, the linear case of Schinzel's Hypothesis H) implies

an affirmative answer for every \(k\). The construction isolates the exact

unproved step: simultaneous primality of an explicit admissible set of \(k\)

linear forms.

The labels used throughout are the requested ones:

[a] elementary-rigorous: a proof is included and uses no unproved

external result.

[b] rigorous-modulo-named-theorem: the claim follows from a named,

stated theorem or a checkable proof-certificate theorem.

[c] plausible/structural-unverified: a heuristic, search miss, or

diagnosis that is not a theorem.

[d] computational-only: the claim is the result of an exact finite

computation, with reproducible code, but is not promoted to an unbounded

theorem.

0. Mandatory live-page gate

[d: live-page observation] I fetched

the live problem page through a

Cloudflare-capable Bright Data browser on 2026-07-26. Direct datacenter HTTP

was not used as authority. The page was last edited 2025-09-28 and displayed

status OPEN.

The verbatim live statement is:

> Let \(k\geq 3\). Are there \(k\) consecutive primes in arithmetic progression?

[d: live-page observation] The page listed references

\([{\rm Er75b}],[{\rm Er83}],[{\rm Er97c}]\), and the following known-result

summary:

Green and Tao proved that there are \(k\) primes in arithmetic progression,

but their primes need not be consecutive.

Erdős described the question as “completely hopeless” in \([{\rm Er83}]\)

and “completely unattackable” in \([{\rm Er97c}]\).

Existence has been verified for \(k\leq 10\).
Even for \(k=3\), infinitude of such progressions remains open.
The page points to Erdős problem 219 and to Guy's problem A6.
It marks the problem as formalised and links OEIS A006560.

[d: live-page observation] The participation/proof markers were:

claimed proofs: 0;
currently working: None;
interested in collaborating: None;
formalisation workers: None;
likes: KStar;
“difficult” marker: KStar;
“tractable” marker: none.

Thus the mandatory stop condition did not fire.

[d: live-page observation] The

linked discussion

contained exactly four comments when fetched; the site itself warns that

comments are unverified:

1. DesmondWeisenberg, 2025-08-11 16:28, sketched a conditional

Hardy--Littlewood prime-tuples construction that also covers the

intermediate positions.

2. Dogmachine, 2025-08-11 16:58, said that Schinzel--Sierpiński (1958) already

yields the conditional implication.

3. Abel, 2026-02-08 23:01, gave a self-contained CRT construction conditional

on Schinzel's Hypothesis H/Dickson's conjecture.

4. Adenwalla, 2026-02-09 06:58, corrected Abel's small-prime congruence:

for \(p\leq k\), the forms are congruent to \(Mn+b\pmod p\), not merely

\(b\pmod p\).

[a] The correction in item 4 does not destroy admissibility: for

\(p\leq k\), all the forms exclude the same one residue class of \(n\), so

\(p-1\) classes remain. Section 4 gives a cleaned-up proof that does not need

the extra small-prime conditions on \(b\).

1. Primary-source and current-state audit

The following claims are deliberately narrower than the sources support.

[b: primary source] Erdős's 1975 paper explicitly asks whether, for

every \(k\), there are \(k\) consecutive primes in arithmetic progression

and says the problem is unattackable even for \(k=3\):

[P. Erdős, Problems and results on combinatorial number theory III,

pp. 295--310](https://www.numdam.org/item/AST_1975__24-25__295_0.pdf).

[b: primary source] Erdős's 1983 paper repeats the conjecture, reports

the then-record six-term example starting at \(121174811\) with difference

\(30\), and again calls it unattackable:

P. Erdős, 1983, PDF.

[b: primary source] Green and Tao prove arbitrarily long arithmetic

progressions of primes, with no consecutiveness assertion:

[B. Green and T. Tao, *The primes contain arbitrarily long arithmetic

progressions*, arXiv:math/0404188](https://arxiv.org/abs/math/0404188),

published in Annals of Mathematics 167 (2008).

[b: primary source] Dubner, Forbes, Lygeros, Mizony, Nelson, and

Zimmermann report CPAPs of lengths \(8,9,10\), found in 1997--1998, and give

the CPAP-10 used below:

[Ten consecutive primes in arithmetic progression, Math. Comp. 71

(2002), 1323--1328](https://doi.org/10.1090/S0025-5718-01-01374-6).

The contemporary

official announcement

gives the same start, step, search parameter, and certification account.

[b: primary source] That 2002 paper explains the divisibility

obstruction, records \(2310\) as the minimum possible difference for

\(k=11\) and \(k=12\), and observes that a CPAP-11 with difference \(2310\)

requires all \(23090\) nonendpoint positions to be composite. It estimated

its then-proposed CPAP-11 search to be at least \(10^{12}\) times the CPAP-10

search and to have an optimum near 900 digits. These are historical search

estimates, not current lower bounds.

[b: bibliographic fact; application proved here] The paper

[A. Schinzel and W. Sierpiński, *Sur certaines hypothèses concernant les

nombres premiers*, Acta Arith. 4 (1958), 185--208](https://eudml.org/doc/206115)

exists and formulates Hypothesis H. I did not treat the forum attribution

as evidence that it spells out this exact CPAP application; Section 4 proves

the implication directly.

[d: maintained record] The

CPAP record table, updated

2026-07-20 when checked, lists examples only through length \(10\).

Its minimal-record page says exact

minima are known only below length \(7\). OEIS

A006560 gives the exact first starts through

length \(6\).

[c: honest search result] Searches of the live tracker, the maintained

record pages, arXiv, and bibliographic/web indexes located no claimed

CPAP-11 and no unconditional post-Green--Tao theorem resolving this

problem. A search miss is not a proof that no unindexed result exists.

2. Elementary structure

Call a length-\(k\) block of consecutive primes a CPAP-\(k\). Write the

primes as

\[ p_i,\ p_{i+1},\ldots,p_{i+k-1}. \]

[a] Equal-gap criterion. This block is an arithmetic progression exactly

when the \(k-1\) adjacent prime gaps

\[ p_{i+1}-p_i,\ldots,p_{i+k-1}-p_{i+k-2} \]

are equal. Consequently an exhaustive ordered prime enumeration is also an

exhaustive CPAP enumeration; no separate choice of subsets is involved.

[a] Primorial obstruction. Suppose

\[ a,\ a+d,\ldots,a+(k-1)d \]

are prime and every term exceeds \(k\). Then

\[ k^\#=\prod_{\substack{q\leq k\\q\ {\rm prime}}}q\mid d. \]

Indeed, if a prime \(q\leq k\) did not divide \(d\), the first \(q\) values

\(a+jd\) would run through every residue class modulo \(q\). One would be

divisible by \(q\), yet would exceed \(q\), contradicting primality. The

hypothesis that the terms exceed \(k\) is necessary: \(3,5,7\), for example,

has step \(2\), not a multiple of \(3\).

[a] Composite burden at minimum step. Put \(d=k^\#\). Between the first

and last endpoints there are

\[ (k-1)d-1 \]

integer positions, of which \(k-2\) are the other AP endpoints. Hence

\[ \#\{\hbox{nonendpoint positions}\}=(k-1)(d-1). \]

This is \(1254\) for \(k=7\), \(1881\) for \(k=10\), and \(23090\) for

\(k=11\). This calculation explains both the published CPAP-10 interval

count and the sharp jump at \(k=11\).

3. A completely checked published CPAP-10

Define \(P\) by the following exact decimal integer:

100996972469714247637786655587969840329509324689190041803603417758904341703348882159067229719

The published progression is

\[ P+210j,\qquad 0\leq j\leq 9, \]

and its exact last term is:

100996972469714247637786655587969840329509324689190041803603417758904341703348882159067231609

[b] Endpoint primality certificate. The companion program embeds the ten

PARI-format Atkin--Morain ECPP chains. For every row

\([N,t,s,a,(x,y)]\), it checks, using integer arithmetic only:

1. \(t^2<4N\), \(s\mid m=N+1-t\), and

\(q=m/s>(N^{1/4}+1)^2\), with the last inequality transformed into an exact

integer inequality;

2. nonsingularity of \(y^2=x^3+ax+b\pmod N\);

3. every denominator used in affine elliptic-curve arithmetic is a unit

modulo \(N\);

4. \(sP\ne\mathcal O\) and \(q(sP)=\mathcal O\);

5. the next row starts at \(q\);

6. the terminal \(q<2^{64}\) is prime by the deterministic seven-base strong

Miller--Rabin theorem with bases

\[ 2,325,9375,28178,450775,9780504,1795265022. \]

These are precisely the conditions in the official

PARI ECPP certificate specification.

Their implication is the Atkin--Morain ECPP primality theorem. For the

terminal test, Theorem 3 of

[Forišek--Jančina, *Fast Primality Testing for Integers That Fit into a

Machine Word*](https://ceur-ws.org/Vol-1326/020-Forisek.pdf)

proves that those seven bases suffice below \(2^{64}\).

[a,b] Consecutiveness certificate. The closed interval has \(1891\)

integers. After removing the ten certified endpoints, the checker proves all

remaining \(1881\) composite: 945 are even, and each of the other 936 has

base \(2\) as an explicit strong Miller--Rabin compositeness witness. Failure

of a strong probable-prime test is a one-sided, rigorous compositeness proof.

Thus the ten certified primes are consecutive.

[d] Integrity checks. The decompressed JSON certificate payload has

SHA-256

e53dbd48f25304b7f8a3232cd662eacc960e4b8baeb4e50f966f8794f4e2269e.

During development, changing the first certificate's \(x\)-coordinate caused

the independent verifier to reject the chain at its elliptic order check.

4. Exact conditional reduction

This section repairs and streamlines the idea in the February 2026 forum

comments.

Fix \(k\geq3\), put

\[ d=k^\#,\qquad S=(k-1)d, \]

and let

\[ R=\{r:1\leq rThus \(R\) is exactly the set of nonendpoint offsets in the desired interval,

and \(|R|=(k-1)(d-1)\).

For every \(r\in R\), choose a different prime \(q_r>S\), and put

\[ M=\prod_{r\in R}q_r. \]

By the Chinese remainder theorem, choose \(b\) satisfying

\[ b\equiv-r\pmod {q_r}\quad(r\in R). \]

Finally define the \(k\) linear forms

\[ L_j(n)=Mn+b+jd,\qquad 0\leq j[a] Forced consecutiveness. For every \(r\in R\),

\[ Mn+b+r\equiv0\pmod {q_r}. \]

For \(n\geq1\), this value exceeds \(q_r\), because \(M>q_r\). Hence every

nonendpoint integer between \(L_0(n)\) and \(L_{k-1}(n)\) is composite.

[a] Admissibility. No prime covers all residue classes of \(n\):

If \(p=q_r\mid M\), then

\(L_j(n)\equiv jd-r\pmod p\). Since \(r\not\equiv0\pmod d\),

\(jd-r\ne0\); and since \(|jd-r|

Thus this \(p\) forbids no \(n\).

If \(p\leq k\), then \(p\mid d\) and \(p\nmid M\). Every form has the

same residue \(Mn+b\pmod p\), so exactly one class of \(n\pmod p\) is

forbidden, leaving \(p-1\).

If \(p>k\) and \(p\nmid M\), each form forbids at most one class, so at most

\(k

This proves that the forms are admissible; it also shows that each form is

primitive and that the forms are distinct.

[b: Dickson/Hypothesis H] Conditional conclusion. Dickson's conjecture

for admissible linear forms says that infinitely many \(n\) make all

\(L_0(n),\ldots,L_{k-1}(n)\) prime. For each such \(n\geq1\), the preceding

forced-compositeness proof says these are \(k\) consecutive primes in

arithmetic progression. Thus Dickson's conjecture gives infinitely many

CPAP-\(k\)'s for every fixed \(k\).

[a,d] Finite arithmetic audit. The checker instantiates the construction

at \(k=5\). It obtains \(d=30\), \(S=120\), 116 forced offsets, distinct CRT

primes from 127 through 839, and 304-digit \(M\) and \(b\). It checks every

CRT congruence and endpoint nonvanishing, the small-prime forbidden-class

counts, and 25 sample primes from the general admissibility case.

[c] Exact remaining lemma. Nothing in the CRT construction proves that

the \(k\) endpoint forms are simultaneously prime even once. The missing

uniform statement is:

> For every \(k\), the explicit admissible family

> \(\{Mn+b+jd:0\leq j

That is already a special case of Dickson's conjecture. A finite search for

some \(n\) can verify a particular \(k\), but it cannot replace this

all-\(k\) lemma.

5. Exact census through \(10^9\)

[d] The checker enumerated every prime \(p\leq10^9\) and scanned all

adjacent equal-gap runs. A CPAP-\(k\) “window” is counted at every starting

prime; consequently, a maximal equal-gap run of \(L\) primes contributes

\(L-k+1\) windows of length \(k\).

|---:|---:|---:|---:|---:|

| 3 | 3 | 2 | 7 | 1,328,401 |

| 4 | 251 | 6 | 269 | 72,131 |

| 5 | 9,843,019 | 30 | 9,843,139 | 85 |

| 6 | 121,174,811 | 30 | 121,174,961 | 1 |

| \(\geq7\) | — | — | — | 0 |

[d] The unique CPAP-6 in the range is

\[ 121174811,\ 121174841,\ 121174871,\ 121174901,\ 121174931,\ 121174961. \]

[d] Counts of maximal equal-gap blocks, by number of primes, were:

| maximal block length | count |

|---:|---:|

| 3 | 1,184,224 |

| 4 | 71,962 |

| 5 | 83 |

| 6 | 1 |

[d] There are 84 maximal blocks of length at least 5 (all printed by the

program). The window/block identity

\[ W_k=\sum_{L\geq k}B_L(L-k+1) \]

was checked for every observed \(k\).

[d] Additional full-stream invariants were

pi(1,000,000,000)                 = 50,847,534
last prime                        = 999,999,937
SHA-256 of 499,999,999 odd flags  = 868e0d150289caba635f86ed92e2c89f47a672595998a69b7869940d46d274d2
sum of primes modulo 2^64         = 24,739,512,092,254,535
xor of primes                     = 6,213,527

[a] Algorithmic completeness. The segmented engine initializes each odd

candidate as prime and crosses out the odd multiples of every prime

\(p\leq\sqrt{10^9}\), starting at

\(\max(p^2,\lceil{\rm low}/p\rceil p)\). Every composite at most the limit

has a prime divisor at most its square root and is therefore crossed out;

starting at \(p^2\) never crosses out a prime. The equal-gap criterion in

Section 2 then makes the CPAP scan exhaustive.

[d] Independent implementation check. A second engine used one global

odd-indexed sieve with a separately derived \(p^2\) index. Separate

\(10^9\) runs produced every same reported statistic and the same complete

flag-stream SHA-256. A --engine both run at \(10^6\) additionally exercised

the program's internal cross-comparison. Equality of SHA-256 fingerprints is

an engineering cross-check, not being presented as a collision-free theorem.

Every reported first occurrence was also rechecked integer-by-integer using

fresh trial division rather than either census flag array.

6. Reproduction

The full executable source, including the certificate data, is

erdos141_wave5j_verify.py. It uses only the

Python 3 standard library and neither downloads a prime table nor calls a

primality package.

Script SHA-256:

56e262fc09382e33fd4d5169ef9f56bb37b38277f98010060ddc6f0f135c9cb6

Run the default exact census plus CPAP-10 proof:

python runs/erdos141_wave5j_verify.py

Run the independently indexed full-array engine:

python runs/erdos141_wave5j_verify.py \
  --limit 1000000000 --engine full --skip-historical-cpap10

Run both engines together on a convenient smoke-test range:

python runs/erdos141_wave5j_verify.py \
  --limit 1000000 --engine both

[d] Measured costs on this x86-64 VM with Python 3.12.3. The final

segmented \(10^9\) run, including all audits and the CPAP-10 proof, took

38.80 seconds and 30,452 KiB maximum resident memory. The separate full-array

\(10^9\) run took 44.36 seconds and 833,588 KiB.

7. What remains and why standard machinery stalls

[a] Green--Tao's theorem does not imply the desired result: it supplies

prime endpoints in arithmetic progression but gives no guarantee that every

integer between adjacent endpoints is composite.

[c] Bounded-gap and prime-cluster methods similarly locate primes at some

offsets in an admissible pattern; they do not simultaneously force all \(k\)

specified AP endpoints prime and exclude every intervening prime. The CRT

handles exclusion exactly, after which the unresolved part is the full

prime-\(k\)-tuples problem rather than another covering congruence.

[c] A larger plain census is not a plausible path to the general

question. At the measured rate, extending the segmented scan to \(10^{12}\)

would cost about 10.8 core-hours under near-linear scaling. The maintained

record page's smallest known CPAP-7 starts near \(7.11\times10^{22}\);

naively scanning that far at the same optimistic rate would cost about

\(7.7\times10^{11}\) core-hours. These are extrapolations, not performance

guarantees. A CPAP-11 search must instead use a specialized CRT/sieving

pipeline over 900-digit-scale candidates; the 2002 paper's \(10^{12}\)

relative-work estimate is too hardware- and algorithm-dependent to turn into

an honest present-day core-hour forecast without first implementing and

benchmarking that search.

[c] The precise wall is therefore mathematical, not merely this run's

CPU budget: prove simultaneous primality for the explicit admissible forms in

Section 4 (even one value for every \(k\)), or introduce a genuinely new

mechanism that avoids that Dickson-strength step. Finite computation can

extend record lengths or bounded nonexistence ranges, but cannot establish

the required uniformity.

PARTIAL: Verified the published CPAP-10 from embedded ECPP certificates, computationally verified that no CPAP-7 ends by 10^9, and reduced the general case conditionally to explicit admissible linear forms; no unconditional k=11 or uniform construction was found.

This is the AI working report, labelled by outcome — not an independently verified claim unless marked PROVED. ← ledger