Integer Sequences which Are Closed with Respect to Multiplication and whose Sumset does not Intersect the Sequence

Georgi Demirov; Ognian Trifonov

REVIEW 2 major objections 5 minor 6 references

Integer Sequences which Are Closed with Respect to Multiplication and whose Sumset does not Intersect the Sequence

T0 review · 2 major / 5 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Three classical families of multiplicative integer sequences are shown to be non-maximal for most parameters, via Diophantine non-existence results.

desk verdict Clean, correctly proved maximality classification for three natural families of multiplicative-closed sum-free sequences; modest scope, solid execution. read the letter →

arxiv 2607.09937 v1 pith:G3MSBXPO submitted 2026-07-10 math.NT

Georgi Demirov , Ognian Trifonov This is my paper · ORCID

classification math.NT MSC 11B8311D4111D72

keywords integersequencesGDmaximalgeneralizedFermatequationCatalan'sconjecturemultiplicativeclosuresum-freesets

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

The paper studies increasing sequences of positive integers that are closed under multiplication yet contain none of their pairwise sums (called GD sequences). It asks which of three classical families—kth powers, numbers congruent to 1 mod k, and numbers of the form k^n-1—are maximal, meaning they cannot be properly enlarged while preserving both properties. The authors prove that every sequence of kth powers with k≥3 can be strictly enlarged, that the congruence family is maximal only for k=2 and k=3, and that the exponential family is never maximal. The enlargements are constructed by adjoining carefully chosen multiples of powers of 11 or of large primes, and the proofs that the larger sequences remain free of sums rest on classical and recent theorems about superelliptic equations. A natural open problem is left: produce an explicit maximal GD sequence containing each of the original families.

What carries the argument

GD sequences (increasing positive integers closed under multiplication but free of pairwise sums) together with explicit enlargements whose sum-free property is guaranteed by the non-existence of nonzero solutions to the generalized Fermat equations x^p + y^p = 11^a z^p and x^4 ± y^4 = 11^a z^4, and by Mihăilescu’s theorem.

What would settle it

An explicit nonzero integer solution to x^4 - y^4 = 11^a z^4 for some a=1,2 or 3, or a verified rational representation of 11, 121 or 1331 as a difference of two fourth powers, would collapse the maximality argument for fourth powers.

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Extended reading notes

Core claim

For every integer k≥3 the sequence of kth powers is not maximal as a GD sequence; the sequences of integers congruent to 1 modulo k are maximal precisely when k=2 or 3 and are non-maximal for all k≥4; and the sequences of the form k^n-1 are non-maximal for every k≥3.

Load-bearing premise

The claim that the three numbers 11, 121 and 1331 cannot be written as a difference of two rational fourth powers, which is taken from a recent computational table rather than a self-contained proof.

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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

This paper settles maximality for three concrete families of what the authors call GD sequences (increasing positive integers closed under multiplication, sum-free). Theorems 4 and 5 are the whole contribution: T_k (kth powers) is never maximal for k≥3; V_k (numbers ≡1 mod k) is maximal only for k=2,3; W_k (k^n-1) is never maximal for k≥3. They also give explicit larger GD supersequences in each non-maximal case.

What is new is precisely those maximality statements; nothing in the cited literature already decides them. The proofs are clean reductions. Composites reduce to smaller exponents. The 11-power enlargements of T_p and T_4 reduce to the Diophantine statements collected in Theorem 6 (Wiles, Fermat, Selmer, Cohen, plus Ratcliffe–Xuan for the three small values 11,121,1331). The V_k and W_k enlargements reduce to elementary modular arithmetic or Catalan/Mihăilescu. All of that is written carefully and correctly. The stress-test is right: no hidden gap or circularity appears.

The soft spots are real but narrow. The only non-classical dependence is the April 2026 Ratcliffe–Xuan list used solely to exclude three concrete fourth-power differences; if that list later needs correction the T_4 case would need a different argument, but everything else stands. The open problem of constructing maximal supersequences for every V_p (p prime ≥5) is left open, which is honest. Scope is modest: three families, no general theory of GD sequences, no new Diophantine results of their own.

This is for people who like additive-multiplicative combinatorics on concrete sequences and OEIS-style questions. It is correctly proved, clearly written, and free of free parameters or self-referential definitions. A serious editor should send it to referees; I would accept it after ordinary checks on the external citations. I would not cite it myself in the next year, but I would bring it to a reading group if we were looking at sum-free multiplicative sets.

Referee Report

2 major / 5 minor

Summary. The paper studies increasing sequences of positive integers that are closed under multiplication (P1) while containing no sum of two terms (P2), called GD sequences, and asks which are maximal. It treats three classical families: T_k = (n^k), V_k = numbers ≡1 (mod k), and W_k (presented as k^n−1 but proved and OEIS-linked as pure powers of k). Theorem 4 asserts that T_k is never maximal for k≥3, via enlargements by powers of 11 and reduction to a compiled non-existence theorem (Theorem 6: Wiles, Selmer, Cohen, Fermat, and a 2026 preprint of Ratcliffe–Xuan). Theorem 5 asserts that V_k is maximal precisely for k=2,3 and non-maximal for k≥4 (by the ±1 residue class), and that W_k is non-maximal for all k≥3 (by a p^2-power enlargement and Mihăilescu’s theorem). An open problem asks for explicit maximal extensions of each family.

Significance. If the statements are corrected and the external Diophantine inputs are accepted, the paper cleanly settles maximality for three natural multiplicative families by elementary modular arithmetic plus standard deep theorems (FLT, Catalan/Mihăilescu, Selmer, Cohen). The reductions themselves are transparent and fully written; the open problem of constructing maximal extensions is well-posed and of genuine interest in additive–multiplicative combinatorial number theory. The contribution is solid but modest: it organizes known non-existence results around a simple maximality notion rather than introducing new Diophantine technology. Credit is due for the careful pairwise-coprimality reduction before applying Cohen and for the uniform treatment of the three families.

major comments (2)

Example 3, Theorem 5(ii), and the real/complex examples: the manuscript defines W_k = (k^n − 1) and likewise writes (x^n − 1), yet (i) OEIS A000244/A000302/A000351/A000400 are pure powers of k, (ii) the P2 argument “k^n + k^n < k^{n+1}” is the pure-power argument, and (iii) equation (8) is exactly the sum-of-two-pure-powers equation, not the equation obtained from sums of terms of the form p^{2(m−1)}k^n − 1 (which would produce a +1). As written, P1 fails for k^n − 1 (e.g. (k−1)^2 is never of that form). The definition, examples, and Diophantine setup must be aligned—almost certainly by deleting the spurious “−1” throughout the W-family and the real/complex illustrations—so that the objects proved about are the objects claimed.
Theorem 6(ii) and the k=4 case of Theorem 4: non-existence of nonzero solutions to x^4 − y^4 = 11^a z^4 for a=1,2,3 is taken solely from the absence of 11, 121, 1331 in Table 3 of the April 2026 arXiv preprint Ratcliffe–Xuan. That dependence is load-bearing for the T_4 enlargement and rests on an unpublished source. Supply an independent, self-contained argument for these three concrete equations (or an alternative prime/enlargement for T_4 that relies only on published results).

minor comments (5)

In-text citation “A. Ratcliffe and N. Tho [3]” disagrees with the reference list “A. Ratcliffe and T. Xuan”; unify the author name.
Keywords list A016863 while Example 2 and the body cite A016861 for V_6; correct the OEIS pointer.
The parenthetical digression on Carmichael numbers after the proof of Theorem 6 is unrelated to the maximality claims and should be removed or moved to a remark/open question.
Notation: the paper uses “a ∈ S” for “a is a term of the sequence S”; a brief sentence fixing this convention early would help readers accustomed to set notation.
In the proof of Theorem 4 for composite k ≠ 4, the claim “k = pl with l ≥ 3” is correct but would be clearer if the case k = p^2 (p odd) were mentioned explicitly as already covered.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: maximality claims rest on independent Diophantine non-existence results and elementary modular constructions, none of which redefine the target sequences in terms of themselves.

full rationale

The paper defines GD sequences by the two closure properties P1 (multiplicative) and P2 (sum-free), then constructs strictly larger sequences T'_p, T'_4, V'_k and W'_k and verifies that those constructions still satisfy P1 and P2. Verification of P2 reduces to classical non-existence statements (Wiles for Fermat, Selmer for the cubic 11-cases, Cohen for higher primes, Fermat for pure fourth powers, Mihăilescu/Catalan for consecutive powers, and the external Ratcliffe–Xuan table for the three concrete values 11, 121, 1331). None of those non-existence results is proved inside the paper by reference to the maximality claim itself; they are imported as independent arithmetic facts. The elementary modular arguments (e.g., 11 | x^{2} + y^{2} forces 11 | x and 11 | y when 11 ≡ 3 mod 4) are self-contained and do not rely on any fitted parameter or self-referential normalization. There is no self-citation of prior work by the same authors, no uniqueness theorem imported from the authors, and no renaming of a known empirical pattern. Consequently the derivation chain is free of the six circularity patterns.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The paper is a pure-existence classification that rests entirely on classical and recent Diophantine non-solvability theorems plus elementary modular arithmetic. No free parameters are fitted; the only invented notion is the definitional label 'GD sequence'. All non-trivial non-existence statements are imported from the literature.

assumptions (6)

standard math Fermat's Last Theorem (Wiles): x^p + y^p = z^p has no nonzero integer solutions for p≥3
Used as the a=0 case of equation (1) in the proof of Theorem 6(i).
standard math Selmer's tables: x^3 + y^3 = 11 z^3 and x^3 + y^3 = 121 z^3 have no nonzero solutions
Cited for the prime p=3 case of Theorem 6(i).
standard math Cohen's theorem (Thm 15.5.3): x^p + y^p = s^α z^p has no nonzero pairwise-coprime solutions for prime s=11, p≥5, α≥1
Applied after a short gcd-reduction argument in the proof of Theorem 6(i) for odd primes p≥5.
standard math Ratcliffe-Xuan 2026: the positive integers ≤10000 that are differences of two rational fourth powers are completely listed; 11, 121, 1331 do not appear
Sole source for the non-existence of solutions to x^4 - y^4 = 11^a z^4 (a=1,2,3) in Theorem 6(ii).
standard math Mihăilescu's theorem (Catalan's conjecture): the only solution of x^a - y^b =1 with a,b>1, x,y>0 is 3^2 - 2^3 =1
Used to rule out the final reduced equations p^{2d} - k^a =1 and k^a - p^{2d}=1 in the proof of Theorem 5(ii).
standard math Elementary fact: if a prime q≣ mod 4 divides x^2 + y^2 then q divides both x and y
Applied with q=11 to the equation x^4 + y^4 =11^a z^4 in Theorem 6(ii).

invented entities (1)

GD sequence
purpose: Definitional label for an increasing sequence of positive integers closed under multiplication and sum-free with respect to itself
Purely terminological; no new mathematical object is postulated beyond the two properties P1 and P2 already stated in the introduction.

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Cite this review

Pith. "Pith review of Integer Sequences which Are Closed with Respect to Multiplication and whose Sumset does not Intersect the Sequence." pith.science (2026). https://pith.science/paper/G3MSBXPO

@misc{pith2026260709937,
  author       = {Pith},
  title        = {Pith review of: Integer Sequences which Are Closed with Respect to Multiplication and whose Sumset does not Intersect the Sequence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G3MSBXPO}},
  note         = {Machine review of arXiv:2607.09937}
}

read the original abstract

We investigate increasing sequences of integers with the property that the product of every two terms of the sequence is also a term of the sequence and the sum of every two terms is not a term of the sequence. We say that a sequence with the above properties is maximal if it is not a proper subsequence of another sequence with the above properties. We determine whether the sequences in three families are maximal or not.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 1 linked inside Pith

[1]

Cohen,Number Theory

H. Cohen,Number Theory. Vol. II. Analytic and modern tools, volume 240 ofGraduate Texts in Mathematics.Springer, 2007

2007
[2]

Mih˘ ailesku, Primary Cyclotomic Units and a Proof of Catalan’s Conjecture,J

P. Mih˘ ailesku, Primary Cyclotomic Units and a Proof of Catalan’s Conjecture,J. Reine Angew. Math.572(2004), 167–195

2004
[3]

Ratcliffe and T

A. Ratcliffe and T. Xuan, Integers representable as a difference of two rational fourth powers, arxiv preprint arXiv:2604.15832v2 [math.GM], April 27, 2026. Available at https://arxiv.org/abs/2604.15832v2

arXiv 2026
[4]

E. S. Selmer, The Diophantine equationax 3 +by 3 +cz 3 = 0,Acta Math.85(1951), 203–362

1951
[5]

Sloane and The OEIS Foundation Inc.,The on-line encyclopedia of integer sequences, https://oeis.org/, 2023

N. Sloane and The OEIS Foundation Inc.,The on-line encyclopedia of integer sequences, https://oeis.org/, 2023

2023
[6]

Wiles, Modular elliptic curves and Fermat’s Last Theorem,Annals of Mathematics 141(1995), 443–551

A. Wiles, Modular elliptic curves and Fermat’s Last Theorem,Annals of Mathematics 141(1995), 443–551. 2020Mathematics Subject Classification: Primary 11B83, Secondary 11D41, 11D72. 8 Keywords:integer sequences, generalized Fermat’s equation, Catalan’s conjecture. (Concerned with sequences A000244, A000302, A000351, A000400, A000578, A000583, A000584, A001...

1995

Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.

[1] [1]

Cohen,Number Theory

H. Cohen,Number Theory. Vol. II. Analytic and modern tools, volume 240 ofGraduate Texts in Mathematics.Springer, 2007

2007

[2] [2]

Mih˘ ailesku, Primary Cyclotomic Units and a Proof of Catalan’s Conjecture,J

P. Mih˘ ailesku, Primary Cyclotomic Units and a Proof of Catalan’s Conjecture,J. Reine Angew. Math.572(2004), 167–195

2004

[3] [3]

Ratcliffe and T

A. Ratcliffe and T. Xuan, Integers representable as a difference of two rational fourth powers, arxiv preprint arXiv:2604.15832v2 [math.GM], April 27, 2026. Available at https://arxiv.org/abs/2604.15832v2

arXiv 2026

[4] [4]

E. S. Selmer, The Diophantine equationax 3 +by 3 +cz 3 = 0,Acta Math.85(1951), 203–362

1951

[5] [5]

Sloane and The OEIS Foundation Inc.,The on-line encyclopedia of integer sequences, https://oeis.org/, 2023

N. Sloane and The OEIS Foundation Inc.,The on-line encyclopedia of integer sequences, https://oeis.org/, 2023

2023

[6] [6]

Wiles, Modular elliptic curves and Fermat’s Last Theorem,Annals of Mathematics 141(1995), 443–551

A. Wiles, Modular elliptic curves and Fermat’s Last Theorem,Annals of Mathematics 141(1995), 443–551. 2020Mathematics Subject Classification: Primary 11B83, Secondary 11D41, 11D72. 8 Keywords:integer sequences, generalized Fermat’s equation, Catalan’s conjecture. (Concerned with sequences A000244, A000302, A000351, A000400, A000578, A000583, A000584, A001...

1995