Sierpinski's Hypothesis H1

Matt Visser (Victoria University of Wellington)

REVIEW 4 major objections 4 minor 10 references

Sierpinski's Hypothesis H1

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Sierpiński's Hypothesis H1—that every row of the n×n array of 1 through n² contains a prime—is verified for every n from 2 to at least 4,553,432,387, with partial guarantees for all larger n.

desk verdict Solid elementary verification of Sierpiński's H1 to 4.5 billion and honest partial results for all n; the math works but the paper needs a cleanup pass on its own numbers and a precise statement of its external-gap dependence. read the letter →

arxiv 2512.22413 v2 pith:PONWITTG submitted 2025-12-27 math.NT

Matt Visser (Victoria University of Wellington) This is my paper · ORCID

classification math.NT MSC 11A4111N05

keywords SierpińskimatricesSierpiński'sHypothesisH1maximalprimegapspigeonholeprincipledistributionprime-countingbounds

topics The Twin Prime Conjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to establish Sierpiński's 1958 Hypothesis H1: when the numbers 1,2,...,n² are arranged in rows of length n, every row contains at least one prime. It proves this for all n up to 4,553,432,387 by combining the known largest prime gap below about 2×10^19, which has width 1,572, with a pigeonhole counting argument: any n consecutive numbers in that range must contain a prime. For arbitrary larger n, it proves that at least one quarter of the rows always contain a prime, and that the bottom 131,294 rows always contain a prime. The interest is that H1 is stronger than two classical prime-gap conjectures, so these bounds sharply constrain how those conjectures could fail.

What carries the argument

The central object is the Sierpiński matrix S_n, whose row i is the n consecutive integers n(n−i)+1,...,n(n−i)+n. The mechanism is a maximum-gap ladder: for a given n, the largest prime gap starting below n² determines, via floor(n/g), a guaranteed minimum number of primes in every row. The proof climbs the ladder: at the top, g=1,572 covers all n from 1,572 to N₈₃; descending through smaller maximal gaps (148, 52, 34, 20, 14, ...) pushes the coverage down to n=13, with direct checks for n≤13. This converts a data table of maximal prime gaps into a universal statement about all rows up to a huge n.

What would settle it

A direct counterexample to the paper's central claim would be a prime-free row in S_n for some n≤4,553,432,387; a finite search over those arrays, or equivalently over all intervals of length n below (4,553,432,387)², would settle it. To test the machinery itself, independently recompute every maximal prime gap starting below (4,553,432,387)²: finding any gap wider than 1,572 would break the pigeonhole step, while confirming the table would corroborate the bound.

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Extended reading notes

Core claim

The load-bearing result is Theorem 1: for all 2≤n≤4,553,432,387, every row of the Sierpiński matrix S_n contains at least one prime. The proof works because the largest prime gap starting below (4,553,432,387)² is the 82nd maximal gap, of width 1,572, so the pigeonhole principle guarantees at least floor(n/1,572) primes in each row for n in that range. The small cases n≤13 are checked directly, or by a known 1961 verification used to cover the overlap n≤4,505; the two ranges splice together to cover all n up to N₈₃. For n beyond N₈₃, Theorems 3 and 7 provide guarantees that at least n/4 rows, and the bottom 131,294 rows, contain primes.

Load-bearing premise

The finite verification succeeds only if the published table of maximal prime gaps up to about 2×10^19 is complete and correct—specifically that no gap wider than 1,572 opens below (4,553,432,387)²—and if the 1961 check really covers n≤4,505.

Editorial extensions

If this is right

If correct, the range 2≤n≤4,553,432,387 verifies the two long-standing interval-prime conjectures that H1 subsumes, because each is a restricted version of the row condition.
The verification is not marginal: for the largest matrix checked, every row contains at least 2,896,585 primes, not just one.
For all larger n, the unconditional one-quarter density guarantee and the 131,294-row bottom guarantee remain true, even though full H1 is open there.
Any future extension of the maximal-gap table will automatically enlarge the finite range, since the proof's only input from the table is the largest gap width below the new endpoint.
A stronger statement than H1 also follows: any n consecutive integers below (4,553,432,387)² contain a prime, not just the special rows of S_n.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The abstract's stated bound of 10,070,368,414 does not match the body's 4,553,432,387; as written, the proof establishes the smaller bound, and the larger figure would need its own defence or correction.
Because the proof converts a single prime-gap width into a statement about all rows, the method transfers directly to any conjecture that demands a prime in every block of n consecutive integers: the data table, not the logic, is the limiting resource.
The one-quarter density result implies that a counterexample to H1 would require a distribution of primes far more uneven than any known evidence: more than three-quarters of the rows of some S_n would have to be prime-free.
A tractable next step would be to run an independent verification of the maximal-gap table up to (4,553,432,387)²; that would upgrade the finite claim from 'as good as the cited table' to a fully self-contained computation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is worth reading and worth refereeing, but only after the author fixes a few admitted inconsistencies. The core result is legitimate: Sierpiński's H1 is verified up to n=4,553,432,387, a huge jump from Schinzel's n≤4505, and the proof is a clean pigeonhole argument using known maximal prime gaps. The partial results for all larger n — at least a quarter of the rows, and the first 131,294 rows — also follow honestly from published Rosser–Schoenfeld, Montgomery–Vaughan, Dusart, and Axler bounds. There are no free parameters, no curve-fitting, and no circularity; the main theorem reduces to the completeness of the maximal-prime-gap table below p*_83≈2.07e19. That table is external, but it is a standard, publicly checked object, and the author cites it. That is an acceptable dependence for a verification paper, provided the provenance is made precise.

The soft spots are real but mostly mechanical. The abstract in the version I saw claims n≤10,070,368,414, while the body proves 4,553,432,387; that discrepancy must be resolved before anything else. Theorem 5's proof says the maximal prime gap in [1,(N_83)^2] is 1676; it is 1572, since the 83rd gap starts above that interval. The bound floor(n/1676) is still valid (weaker than the true floor(n/1572)), so Theorem 5 survives, but the statement as written is false. Table I contains at least one incompatible entry (the 4.5e9 row lists 1752 instead of 1572), and the descent step has notation typos (g*_21 for g*_13, etc.). These are not hard to fix, but they are exactly the kind of thing that prevents clean acceptance.

The only substantive worry is the completeness of the maximal-gap table. If an undiscovered gap of size >1572 hides below p*_83, the main finite verification breaks. I don't think that is likely, but the paper should state explicitly that the proof depends on the table being complete up to that point, and ideally point to the exhaustive-search verification rather than a Wikipedia page. The author does not ship the gap data or an independent check, which is normal for a short note, but worth flagging.

Bottom line: serious referee, conditional on the author cleaning up the numerical errors and the abstract/body mismatch. The central argument holds up as far as I can tell, and the verification is a real, if limited, advance.

Referee Report

4 major / 4 minor

Summary. The paper studies Sierpiński's Hypothesis H1, the conjecture that for n≥2 each row of the n×n array containing 1,…,n² has a prime. The main theorem (Theorem 1, §3.1) asserts that H1 holds for all n up to N83 = 4,553,432,387. The proof uses the known maximal prime gaps: since the largest gap below p*83 is claimed to be g*82 = 1572, every block of 1572 consecutive integers below N83² contains a prime, so each row of S_n with n≥1572 has at least floor(n/1572)≥1 prime; the small range n≤4505 is covered by Schinzel's result (or by the descent argument in §3.2). Additional results: Theorem 3 shows at least n/4 rows of S_n contain a prime for any n; Theorem 7 shows the first 131,294 rows contain a prime for n>N83; Theorem 6 gives a descent property. The paper relies on external maximal-gap tables and explicit Chebyshev bounds.

Significance. If the main theorem is sound, it is a substantial numerical verification of H1, extending the known range by roughly nine orders of magnitude and implying Oppermann's and Legendre's conjectures up to 4.5×10⁹. The proof is elementary and transparent, and the descent argument (§3.2) is a self-contained way to cover the range once the gap data are supplied. The partial results — the quarter-of-rows bound and the lower-row bound — are original and potentially useful. However, the claim is conditional on the completeness of the external maximal-prime-gap table up to p*83, and the manuscript contains several internal numerical inconsistencies that must be fixed; therefore the current version is not yet publishable.

major comments (4)

[Abstract] The arXiv abstract states that H1 is verified for n≤10,070,368,414 and that partial results hold for n≥10,070,368,414. The body and the paper's own abstract prove bounds with 4,553,432,387 (and n≥4,553,432,388). This is a factor of about 2.2 overclaim and must be corrected in the metadata; as written it is inconsistent with Theorem 1.
[§3.1, Theorem 1] The proof hinges on the assertion that the largest prime gap with starting point below p*83 is g*82 = 1572, so every block of 1572 consecutive integers below N83² contains a prime. This is equivalent to the completeness of the external table of maximal gaps up to p*83 ≈ 2.07×10^19. The paper cites Wikipedia, PrimePages, and primecount but provides no data, certificate, or reproducible verification. If a gap >1572 with starting point <p*83 existed, the pigeonhole argument would fail for some n≤N83. This is a load-bearing external assumption; the manuscript should either supply the verified table (or code to reproduce it) or explicitly state the theorem as conditional on this computational record.
[§4.2, Theorem 5] The proof states: 'We have already seen that the maximal prime gap in the interval [1,(N83)^2] is 1676.' This is incorrect: the 83rd maximal gap of size 1676 starts at p*83, which is greater than N83²; the largest gap inside the interval is g*82 = 1572. The conclusion of Theorem 5 remains true if 1572 is used (since n>N83 makes floor(n/1572)≥1), but the proof as written relies on a false premise.
[§3.3, Table 1] Table 1 is internally inconsistent. For n=4.5×10^9 the table gives i_min=83 and g*_{i_min-1}=1752, but g*82 is 1572, and a value 1752 larger than the 83rd gap of 1676 is impossible. Similarly the entry for n=4×10^9 gives i_min=79 and g*78=1525, which appears to be off by one or more indices; the #min column in several rows seems to be computed with different gap values. The table should be recomputed and checked against a reliable source.

minor comments (4)

[§3.2, Step 2] The text states floor[1571/148] = 11; the correct value is 10. This does not affect the covering argument, but it is a factual error. Also, in Step 3 the labels g*21 and n*22 should be g*13 and n*14 respectively.
[§3.2, Step 7] The text says (13)² = 167; it should be 169.
[§4.3, Theorem 7] The intervals k∈[1,15957], [2,54957], and [5,131294] are stated to be 'easily checked', but the numerical check is not shown. A short table or script reproducing the inequality (4.20) would improve verifiability.
[§2, Lemma (4.4)] The equivalence in (4.4) uses k≥1, omitting the bottom row k=0. The bottom row is trivial for n≥2, but the statement should say so explicitly to avoid a logical gap.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: proof uses independent external prime-gap and explicit-bound data; no self-citation, fitted prediction, or definitional chaining.

full rationale

Walking the derivation chain: Theorem 1 (§3.1) reduces H1 for n∈[1572,N83] to the assertion that every interval of length 1572 below p*83 contains a prime, importing the maximal prime gap table (refs [4-6]) as an external, independently computed benchmark. Schinzel's n≤4505 result covers the overlap. Theorem 7 uses explicit published bounds of Dusart and Axler, and the Montgomery-Vaughan large sieve bound and Rosser-Schoenfeld estimate are standard external theorems. No parameter is fitted to H1 data; the 'some experimentation' is a choice among published constants. No definition of H1 is defined in terms of maximal gaps, and the maximal gaps do not depend on H1. The paper contains no self-citations by the author, no uniqueness theorem imported from prior author work, and no renaming of a known result. The skeptical concern about completeness of the gap table is a correctness/verification risk, not a circularity: an incomplete table would make the proof have a false premise, but the premise is not derived from the conclusion. Hence no circular step, score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. All external inputs are published computational tables and analytic bounds. The central finite verification depends heavily on the completeness of the maximal prime gap table, which is an external dataset assumption rather than a fitted parameter.

assumptions (6)

domain assumption The known maximal prime gap table up to p*_83≈2.07×10^19 is complete and correct; in particular the largest prime gap below N_83² is the 82nd maximal gap of size 1572.
Invoked in §3.1 to assert that every block of n≥1572 consecutive integers below n² contains a prime. The paper cites Wikipedia and the PrimePages table, but does not independently verify the data.
domain assumption Schinzel's 1961 result that Sierpiński's H1 holds for all n∈[2,4505] is correct.
Used in §3.1 to cover the small-n region not reached by the maximal gap argument. Cited as reference [2].
standard math π(x)>x/ln x for real x≥17 (Rosser–Schoenfeld)
Used in §4.1 to lower-bound the total number of primes in the first n² numbers.
standard math Montgomery–Vaughan large sieve bound π(x+y)<π(x)+2y/ln y for the relevant ranges.
Used in §4.1 to upper-bound the number of primes in any single row, enabling the quarter-of-rows pigeonhole result.
domain assumption Explicit Chebyshev bounds |ϑ(x)-x|<η_m x / ln^m x for x>x_m with the three parameter choices from Dusart and Axler.
Used in §4.3 to guarantee primes in the first 131,294 rows. The constants are from published sources, not proved in this paper.
domain assumption Breusch's result that for x≥48 there is a prime between x and (9/8)x, used via the (8n,9n) interval to cover the lowest rows.
Invoked in §1 and §4.3 for the first few rows near the bottom of the matrix.

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Cite this review

Pith. "Pith review of Sierpinski's Hypothesis H1." pith.science (2026). https://pith.science/paper/PONWITTG

@misc{pith2026251222413,
  author       = {Pith},
  title        = {Pith review of: Sierpinski's Hypothesis H1},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PONWITTG}},
  note         = {Machine review of arXiv:2512.22413}
}

abstract

Sierpinski's Hypothesis H1, formulated in 1958, is the conjecture that (provided $n\geq 2$), when the first $n^2$ counting numbers, $1, 2,3,\dots n^2$, are arranged in a square, then each row contains at least one prime. This conjecture is particularly interesting in that it subsumes and is stronger than both the Oppermann and Legrendre conjectures. Herein I shall verify Sierpinski's Hypothesis H1 for (at least) the first $n \leq \hbox{10 070 368 414} \approx 10 \hbox{ billion}$ of these Sierpinski matrices. I shall also demonstrate some partial but more general results. For example: Even for arbitrary $n\geq \hbox{10 070 368 414}$ at least one quarter of the rows of the $n$th Sierpinski matrix contain at least one prime. Furthermore, even for arbitrary $n\geq \hbox{10 070 368 414}$ at least the first $\hbox{141 618}$ rows of the $n$th Sierpinski matrix always contain at least one prime. These and related results are obtained largely by using the locations and values of the known maximal prime gaps, the pigeonhole principle, and some recent bounds on the first Chebyshev function.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 2 canonical work pages

[1]

Sur certaines hypoth` eses concernant les nombres premiers

Andrejz Schinzel and Waclaw Sierpi´ nski “Sur certaines hypoth` eses concernant les nombres premiers”, (On certain hypotheses concerning the prime numbers), Acta Arithmetica4 # 3(1958) 185–208

1958
[2]

Remarks on the paper “Sur certaines hypoth` eses concernant les nombres premiers

Andrejz Schinzel, “Remarks on the paper “Sur certaines hypoth` eses concernant les nombres premiers””, Acta Arithmetica7 # 1(1961) 1–8

1961
[3]

Zur Veralgemeinerung des Bertrandschen Postulates, das zischenxund 2xstets Primeahlen liegen

R. Breusch, “Zur Veralgemeinerung des Bertrandschen Postulates, das zischenxund 2xstets Primeahlen liegen”, (Generalizing Bertrand’s postulate that there are always prime numbers betweenxand 2x), Mathematische Zeitschrift34(1932) 505–526

1932
[4]

Prime gap

Wikipedia, “Prime gap”, https://en.wikipedia.org/wiki/Prime gap (accessed on 27 December 2025)

2025
[5]

Prime pages, Table of known maximal gaps, https://t5k.org/notes/GapsTable.html (accessed on 27 December 2025)

2025
[6]

Kim Walisch, 2025,primecountsoftware, https://github.com/kimwalisch/primecount

2025
[7]

Approximate formulas for some functions of prime numbers

J. Barkley Rosser and Lowell Schoenfeld, “Approximate formulas for some functions of prime numbers”, Illinois J. Math.6 #1(March 1962) 64–94. doi:10.1215/ijm/1255631807

arXiv 1962
[8]

The large sieve

H. L. Montgomery and R. C. Vaughan, “The large sieve”, Mathematika20, Part 2 # 40(1973), 119–134. doi:10.1112/S0025579300004708

work page doi:10.1112/s0025579300004708 1973

Show all 10 references

[9]

Explicit estimates of some functions over primes

Pierre Dusart, “Explicit estimates of some functions over primes”, Ramanujan Journal45 #1(2018) 227–251. doi:10.1007/s11139-016-9839-4

2018 doi
[10]

New estimates for some functions defined over primes

Christian Axler, “New estimates for some functions defined over primes”, Integers18(2018) #52https://math.colgate.edu/ ~integers/s52/s52.pdf [arXiv: 1703.08032 [math.NT]] – 14 –

2018 arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Sur certaines hypoth` eses concernant les nombres premiers

Andrejz Schinzel and Waclaw Sierpi´ nski “Sur certaines hypoth` eses concernant les nombres premiers”, (On certain hypotheses concerning the prime numbers), Acta Arithmetica4 # 3(1958) 185–208

1958

[2] [2]

Remarks on the paper “Sur certaines hypoth` eses concernant les nombres premiers

Andrejz Schinzel, “Remarks on the paper “Sur certaines hypoth` eses concernant les nombres premiers””, Acta Arithmetica7 # 1(1961) 1–8

1961

[3] [3]

Zur Veralgemeinerung des Bertrandschen Postulates, das zischenxund 2xstets Primeahlen liegen

R. Breusch, “Zur Veralgemeinerung des Bertrandschen Postulates, das zischenxund 2xstets Primeahlen liegen”, (Generalizing Bertrand’s postulate that there are always prime numbers betweenxand 2x), Mathematische Zeitschrift34(1932) 505–526

1932

[4] [4]

Prime gap

Wikipedia, “Prime gap”, https://en.wikipedia.org/wiki/Prime gap (accessed on 27 December 2025)

2025

[5] [5]

Prime pages, Table of known maximal gaps, https://t5k.org/notes/GapsTable.html (accessed on 27 December 2025)

2025

[6] [6]

Kim Walisch, 2025,primecountsoftware, https://github.com/kimwalisch/primecount

2025

[7] [7]

Approximate formulas for some functions of prime numbers

J. Barkley Rosser and Lowell Schoenfeld, “Approximate formulas for some functions of prime numbers”, Illinois J. Math.6 #1(March 1962) 64–94. doi:10.1215/ijm/1255631807

arXiv 1962

[8] [8]

The large sieve

H. L. Montgomery and R. C. Vaughan, “The large sieve”, Mathematika20, Part 2 # 40(1973), 119–134. doi:10.1112/S0025579300004708

work page doi:10.1112/s0025579300004708 1973