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.

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

yields the conditional implication.

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

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

1. 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.

• [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, 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. 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 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

[a] Equal-gap criterion. This block is an arithmetic progression exactly when the \(k-1\) adjacent prime gaps

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

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

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

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

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

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;

1. nonsingularity of \(y^2=x^3+ax+b\pmod N\);
2. every denominator used in affine elliptic-curve arithmetic is a unit

modulo \(N\);

1. \(sP\ne\mathcal O\) and \(q(sP)=\mathcal O\);
2. the next row starts at \(q\);
3. 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 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

and let

Thus \(R\) 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

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

Finally define the \(k\) linear forms

[a] Forced consecutiveness. For every \(r\in R\),

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 \(|jd-r|<S<p\), it is also nonzero modulo \(p\). 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

\(k<p\) of the \(p\) classes are forbidden.

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 \(\{Mn+b+jd:0\leq j<k\}\) constructed above has a simultaneous prime value.

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 \(k\).

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

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

[d] There are 84 maximal blocks of length at least 5 (all printed by the program). The window/block identity

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.