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REVIEW 4 major objections 5 minor 3 references

A New Primes-Generating Sequence

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A new gcd-filtered quadratic generates only 1's and primes, more densely than Rowland's sequence.

desk verdict The real mathematical claim is true, but the paper hides the proof and overstates what is shown; worth a serious referee if only to make the author write the missing argument. read the letter →

arxiv 2509.09745 v1 pith:RNDN4KQ6 submitted 2025-09-11 math.GM

Mohammed Bouras This is my paper · ORCID
classification math.GM MSC 11A4111B8311A55
keywords prime-generatingsequencegcdrecurrenceRowlandcontinuedfractionquadraticpolynomialleftfactorialintegernumbertheory
topics The Collatz 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

This paper introduces a new integer sequence defined by dividing a quadratic polynomial by its greatest common divisor with an auxiliary recursively defined term. The author claims that every term is either 1 or a prime, that the sequence produces primes more efficiently than Rowland's well-known prime-generating recurrence, and that all primes ending in 1 or 9 appear. A continued-fraction identity connects the auxiliary recurrence to a simpler factorial form, and symmetry properties are conjectured for repeated primes. The supporting evidence is computational up to 10,000 terms, and a key simplification is asserted without proof.

What carries the argument

The core mechanism is the gcd filter: a quadratic numerator x = n^2 − n − 1 is divided by gcd(x, y), where y = b(n−3) + n b(n−4). The auxiliary sequence b is defined by a two-term linear recurrence with polynomial coefficients. The unproved identity y ≡ (n−1)! (in the sense of equal gcds) is the algebraic link that turns the denominator into a factorial-like filter. Theorem 1 establishes a finite continued-fraction identity involving b and m, which is used to generalize the construction to a family of quadratic and linear forms.

What would settle it

Compute a(n) directly from the recurrence b(n) for all n up to, say, 100,000 and look for any composite a(n) > 1. Alternatively, compare gcd(n^2−n−1, b(n−3)+n b(n−4)) with gcd(n^2−n−1, (n−1)!) for the same range; any mismatch would falsify the unproved simplification and, with it, the structural explanation of the sequence.

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Extended reading notes

Core claim

The central claim is that for n ≥ 3, a(n) = (n^2 − n − 1)/gcd(n^2 − n − 1, b(n−3) + n b(n−4)) takes only the value 1 or prime values, where b(n) obeys b(n) = (n+2)(b(n−1) − b(n−2)) with b(−1)=0, b(0)=1. The paper further states that b(n−3) + n b(n−4) can be replaced by (n−1)! in the gcd without changing the result, yielding a finite continued fraction representation and a connection to the left factorial function. The sequence is claimed to be a denser prime source than Rowland's recurrence, and a symmetry conjecture predicts that each prime (except 5) appears exactly twice at positions related to p.

Load-bearing premise

The paper's structural conclusions rely on the unproved statement that b(n−3) + n b(n−4) shares the same gcd with n^2−n−1 as (n−1)! does; if this identity fails for some n, the sequence's prime-only behavior and the continued-fraction claims are not established.

Editorial extensions

If this is right

  • If correct, the sequence provides a new explicit prime-generating formula: among the first 10,000 terms, 8,580 are primes and only 1,420 are the value 1, a denser prime output than Rowland's recurrence.
  • All primes ending in 1 or 9 would appear in the sequence, covering two complete residue classes of primes.
  • The symmetry property a(p−n+1)=p would allow the positions of repeated primes to be predicted directly from p, potentially simplifying the search for large primes.
  • The continued-fraction representation gives an alternative way to compute the denominator without iterating the b recurrence, and connects the sequence to the left factorial function.
  • The generalized family a_k(n) extends the construction to several quadratic and linear forms, each mostly producing 1's and primes, though one observed exception (a_4(n)=4) shows the property is not universal.

Reading between the lines

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

  • The unproved factorial identity may be a special case of a broader congruence between linear recurrences with polynomial coefficients and factorial functions; establishing it could unify this sequence with other gcd-based prime generators.
  • The claimed efficiency (only 14.2% non-primes) rests on a finite computational range; a natural extension is to investigate the asymptotic density of 1's and compare it rigorously with Rowland's sequence.
  • The symmetry conjecture suggests an involution on indices n ↔ p−n+1; understanding this structure might connect the sequence to prime-counting functions or modular properties of the quadratic form.
  • The fragility of the prime-only property under parameter changes (as seen with a_4(n)=4) raises the question of which k values preserve the property and whether the factorial identity is the precise condition.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper defines an integer sequence a(n) for n≥3 by a(n) = (n^2 − n − 1)/gcd(n^2 − n − 1, b(n−3) + n b(n−4)), where b(n) = (n+2)(b(n−1) − b(n−2)) with b(−1)=0, b(0)=1. The abstract asserts that a(n) contains only 1's and primes, that it has a finite continued fraction representation, and that it is 'more efficient' than Rowland's prime-generating recurrence. Section 2, however, labels the 1-or-prime property as Conjecture 1, supported only by a 10,000-term computational check. The paper also presents two continued fraction identities (Theorems 1 and 2) and a family of generalized sequences a_k(n).

Significance. If the 1-or-prime property is established, this construction gives a compact and notably dense prime-producing sequence with a simple quadratic numerator, and the conjectured symmetry a(p−n+1)=p is interesting and easily testable. The manuscript verifies the first 10,000 terms and links the sequence to OEIS entries (A356247 and related), which is useful. However, as it stands the paper does not prove its central claim, and the continued-fraction theorems contain serious gaps and apparent errors. The value of the submission is currently that of a well-documented conjecture rather than a proved theorem.

major comments (4)
  1. [Abstract and §2, Conjecture 1] The abstract states as a finding that a(n) contains only 1's and primes, but §2 explicitly formulates this as Conjecture 1 and offers only a 10,000-term computation. Since this is the central claim of the paper, the abstract must be corrected to describe the statement as conjectural, or a proof must be supplied.
  2. [§1, gcd replacement] The sentence 'It is immediate to observe that the combination b(n−3)+n b(n−4) can be replaced by (n−1)! in the greatest common divisor without altering the result' is the load-bearing simplification behind the §2 case analysis. No proof or reference is given. This gap must be closed. A proof can be supplied from the closed form for b(n) via the congruence 2(n−2)(b(n−3)+n b(n−4)) ≡ (n−1)! (mod n^2−n−1), but the manuscript does not contain it.
  3. [§3, Theorems 1 and 2] The proofs are not rigorous and the statements contain apparent errors. In Theorem 1's proof, Eq. (4) is algebraically wrong: for n=4 the recurrence gives a_1=3a_3−8a_4, not 3a_3−14a_4; the correct coefficient appears to be −n(n−2). Theorem 2 as stated fails for small n: for n=3 the left-hand side is 2m/(3m−6), while the right-hand side simplifies to 2(m−1)/(m−2). These identities need to be restated precisely and proved before the finite-continued-fraction claim in the abstract can be accepted.
  4. [§1 and Abstract, efficiency claim] The assertion that the sequence is 'more efficient for generating prime numbers than the Rowland sequence' is never defined or quantified. Specify the metric (e.g., proportion of non-1 terms that are prime, or density of primes up to N) and provide a numerical comparison. Without this, the claim is not falsifiable.
minor comments (5)
  1. [§1] The text says 'Where b(n)=...' should be 'where b(n)=...'.
  2. [§2] The three cases listed do not explicitly exclude the situation gcd(x,y)=1 with x composite. The text should note that Conjecture 1 asserts that this case does not occur.
  3. [§4, after Table 2] The sentence 'For 1≤k≤5, all terms in the sequence a_k(n)...' refers to the linear family in Table 2, not the quadratic family in Table 1. State this explicitly, since the k=3 row of Table 1 contains the composite value 9 and otherwise the claim appears contradictory.
  4. [§3] The parameter m in Theorems 1 and 2 is not defined; specify that it is a positive integer and the range of n for which the identities are intended.
  5. [§1] When listing the first sequence terms, state clearly that the indexing starts at n=3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central 1-or-prime claim is a genuine (though underproved) number-theoretic consequence of the defining gcd, not a fitted parameter or self-citation chain.

full rationale

Walking the derivation chain: the sequence is defined by a(n) = x / gcd(x, b(n-3)+n b(n-4)), x = n^2 - n - 1. The key load-bearing assertion is the unproved statement in Section 1 that b(n-3)+n b(n-4) can be replaced by (n-1)! inside the gcd. That assertion is not circular: it is a mathematical identity about the auxiliary recurrence, not a definition of a(n) in terms of the target property. If supplied, the identity would make the 1-or-prime claim a standard consequence: since x < n^2, gcd(x,(n-1)!) removes all prime factors below n, leaving at most one prime factor. The paper omits the proof, so there is a serious correctness gap, but the gap is not a reduction of the conclusion to its own premise. Theorem 1's proof is sketchy ('comparing' and 'by tracing the recurrence'), but it is an attempt to derive a continued-fraction identity from the recurrence, not a self-citation or a renamed fit. The OEIS references are labels, not load-bearing citations, and no uniqueness theorem or prior work by the authors is invoked to force the construction. The 'more efficient than Rowland' claim is unquantified and unsupported, but that is an overclaim, not circularity. Hence no circular step is exhibited.

Assumptions & free parameters 2 free parameters · 2 assumptions · 0 invented entities

The central claim rests on an unproved simplification identity and an unproved inductive extrapolation from 10,000 computed terms. The hand-picked initial conditions and quadratic are ad hoc choices rather than derived parameters.

free parameters (2)
  • Initial conditions b(-1)=0, b(0)=1 = 0; 1
    Chosen by hand; the entire b(n) sequence and hence every a(n) depends on these two values, with no stated reason for why they yield only primes.
  • Quadratic polynomial n^2 - n - 1 = coefficient of n: -1; constant: -1
    Ad hoc choice; the prime-producing behavior is only observed for this polynomial, and the paper does not derive it from a principle. The generalized family in Section 4 uses similar hand-picked polynomials.
assumptions (2)
  • domain assumption gcd(n^2-n-1, b(n-3)+n b(n-4)) = gcd(n^2-n-1, (n-1)!) for all n
    Stated as 'It is immediate to observe' in Section 1 with no proof; the simplicity and the claimed filter mechanism depend on it.
  • domain assumption The pattern of terms being 1 or prime observed up to n=10,000 continues for all n
    This is Conjecture 1 in Section 2; the abstract's factual wording depends on this unproved extrapolation.

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Cite this review

Pith. "Pith review of A New Primes-Generating Sequence." pith.science (2026). https://pith.science/paper/RNDN4KQ6

@misc{pith2026250909745,
  author       = {Pith},
  title        = {Pith review of: A New Primes-Generating Sequence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RNDN4KQ6}},
  note         = {Machine review of arXiv:2509.09745}
}
abstract

For the sequence defined by \[ a(n) = \frac{n^2 - n - 1}{\gcd\big(n^2 - n - 1,\, b(n-3) + n\,b(n-4)\big)} \] Where $b(n) = (n+2)\big(b(n-1) - b(n-2)\big),$ with initial conditions $b(-1) = 0$ and $b(0) = 1$, we find that $a(n)$ contains only $1$'s and primes, and can be represented as a finite continued fraction. It is more efficient for generating prime numbers than the Rowland sequence.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

3 extracted references · 1 linked inside Pith

  1. [1]

    Rowland,A Natural Prime-Generating Recurrence, Journal of Integer Se- quences, Vol

    Eric S. Rowland,A Natural Prime-Generating Recurrence, Journal of Integer Se- quences, Vol. 11 (2008)

    2008

  2. [2]

    Benoit Cloitre,10 conjectures in additive number theory, arXiv preprint arXiv:1101.4274 (2011), https://arxiv.org/abs/1101.4274

    arXiv 2011

  3. [3]

    N. J. A. Sloane,The On-line Encyclopedia of Integer Sequences,https://oeis.org. 5

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Reviewed August 4, 2026 · model on record in the stance chip above.