More on consecutive multiplicatively dependent triples of integers

Ingrid Vukusic; Istv\'an Pink; Michael A. Bennett

REVIEW 3 major objections 3 minor 1 cited by

More on consecutive multiplicatively dependent triples of integers

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A new theorem completely classifies integer triples (2,b,c) for which (2,b,c), (3,b+1,c+1), and (4,b+2,c+2) are each multiplicatively dependent.

desk verdict Theorem 1.3 is false as stated: (2,3,8) and (2,6,8) are missing, and the listed exception (2,6,16) is not a solution; the Section 4 restatement is also inconsistent. read the letter →

arxiv 2411.12009 v2 pith:YRKQHTIR submitted 2024-11-18 math.NT

Michael A. Bennett , István Pink , Ingrid Vukusic This is my paper · ORCID

classification math.NT MSC 11N2511D6111J86

keywords multiplicativedependenceconsecutivetriplesexponentialDiophantineequationslinearformsinlogarithmslatticereductionx^2-2=y^n

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

An integer triple is multiplicatively dependent when some nontrivial choice of integer exponents makes the product of its entries equal to 1; for example, $(2,4,14)$ is one because $2^2\cdot 4^{-1}=1$. The paper proves a complete answer for triples of the form $(2,b,c)$ in which $(2,b,c)$, $(3,b+1,c+1)$, and $(4,b+2,c+2)$ are each multiplicatively dependent: every such triple lies in one of two explicit infinite families, or is one of the two isolated cases $(2,4,14)$ and $(2,6,16)$. A second theorem classifies the stricter variant in which each dependence already appears inside a two-entry subtriple, yielding exactly the shapes $(2,6,8)$, $(2,8,2^x-2)$, and $(2,8,10^y-2)$. This matters because earlier work had shown that for every fixed first entry $a\ge 3$ there are only finitely many examples, while $a=2$ was known to have infinite families; the paper pins down precisely what happens in that remaining case.

What carries the argument

The carrying object is the reduction of multiplicative dependence to exponential Diophantine equations. After writing each entry as a power of a fixed base times a remaining factor, the three dependences are forced into equations of the shape $2^y(2^x\pm1)^z\mp1=3^r(2^{x+1}\pm1)^w$. The solution method is layered: elementary valuation arguments and classical results on perfect powers remove easy solution families; an explicit lower bound for linear forms in logarithms bounds all exponents; lattice reduction shrinks the range to $3\le x\le 296$; and a final computer check over the residual triples yields exactly the listed solutions. For Theorem 1.7 the same reduction produces the three equations of Problem 1.6, and their non-solvability is proved using strong parametric bounds for solutions of $x^2-2=y^n$, including a proof that any exponent $n$ in that equation lies between 41 and 1237.

What would settle it

Recompute the finite verification steps independently: for every $3\le x\le 296$, test all pairs $(y,z)$ satisfying the reduced bound $y+(x-0.2)z-1<U(x)$ in equations (3.22) and (3.23); any solution outside the listed families disproves Propositions 3.1 and 3.2 and hence Theorem 1.3. Likewise recompute the factorizations of $2^{t-1}-1$ for the 102 primes $t$ with $41\le t\le 1237$ and $t\equiv 13,17,19,23\pmod{24}$, and test every divisor of the form $d^x$ with $x\ge 2$ for the congruence $d^x\equiv 1\pmod Q$; a hit would falsify Proposition 6.3.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1.3: if $2<b<c$ and each of $(2,b,c)$, $(3,b+1,c+1)$, $(4,b+2,c+2)$ is multiplicatively dependent, then $(2,b,c)$ is one of the parametric families $(2,8,2^x5^y-2)$ or $(2,2^x-2,2^{2x}-2^{x+1})$, or one of the two exceptional triples $(2,4,14)$, $(2,6,16)$. The classification is exhaustive: the proof splits into parity cases, shows that one of $b,c,b+2,c+2$ must be a power of two, and then reduces the remaining alternatives to the two exponential equations $2^y(2^x+1)^z-1=3^r(2^{x+1}+1)^w$ and $2^y(2^x-1)^z+1=3^r(2^{x+1}-1)^w$, whose complete solution sets are determined in Propositions 3.1 and 3.2. The paper also proves Theorem 1.7, the corresponding classification for triples that are 2-multiplicatively dependent at all three steps, with the three parameterized families just listed.

Load-bearing premise

The classification depends on the completeness of finite computer checks: the residual check for $3\le x\le 296$ in the two exponential equations, and the factorizations of $2^{t-1}-1$ for the 102 relevant primes sourced from an external database; if any check or factorization is wrong or incomplete, an unlisted solution could survive.

Editorial extensions

If this is right

The $a=2$ case is now closed: the two parametric families $(2,8,2^x5^y-2)$ and $(2,2^x-2,2^{2x}-2^{x+1})$ account for all infinite behaviour, and $(2,4,14)$, $(2,6,16)$ are the only sporadic triples.
Combined with the earlier finiteness theorem for $a\ge 3$, this gives a complete dichotomy for fixed first entry: finitely many triples for every $a\ge 3$, and exactly the two families plus two exceptions for $a=2$.
The three Diophantine equations of Problem 1.6 have no solutions at all, so the only consecutive 2-multiplicatively dependent triples are $(2,6,8)$, $(2,8,2^x-2)$ with $x\ge 4$, and $(2,8,10^y-2)$ with $y\ge 2$.
The proof yields an effective bound $n\le 1237$ for exponents in $x^2-2=y^n$, a standalone contribution to an equation that remains open in general.
Any triple $(2,b,c)$ satisfying the three consecutive dependencies must have one of $b,c,b+2,c+2$ equal to a power of two, a structural constraint of independent use.

Reading between the lines

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

Not stated in the paper, the two families are sparse: family $(2,8,2^x5^y-2)$ has $O((\log X)^2)$ triples with $c\le X$ and family $(2,2^x-2,2^{2x}-2^{x+1})$ has $O(\log X)$, so the total count grows only polylogarithmically.
The same equation-solving pipeline should make the finitely many $a\ge 3$ cases of the earlier finiteness theorem computationally explicit; the paper does not do that, but its bounds indicate the range is tractable.
Theorem 1.7, combined with Theorem 1.3, implies that any infinite family answering the open question about consecutive multiplicatively dependent triples would have to come from varying the first entry $a$ rather than from a fixed $a$; an implicit but direct consequence.
A fully independent recomputation of the finite factorization checks would remove reliance on the external database; the paper's described verification procedure is precise enough to be rerun.

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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

Colleague — you should know upfront: the main classification theorem is wrong as printed. The triple (2,3,8) satisfies all three multiplicative dependence conditions (2^3=8, 3^2=9, 4·5^2=10^2), and it is not in family (1.1), family (1.2), or the listed exceptions. Same for (2,6,8). Meanwhile the listed exception (2,6,16) fails: (3,7,17) is a triple of distinct primes, hence multiplicatively independent. Section 4's own restatement of Theorem 1.3 is closer—it includes (2,3,8) and (2,6,8)—but it also includes (2,4,8), which fails at (4,6,10), and it still keeps (2,6,16). So the theorem statement and the proof's summary are inconsistent, and both contain invalid entries.

That's a load-bearing flaw because Theorem 1.3 is the headline result. But I want to give the paper its due. The machinery is substantial: they solve the two exponential equations (1.3) via Matveev, Laurent, and LLL, with SageMath code linked; they prove Proposition 5.1 (bounds on the Lebesgue–Nagell equation x^2−2=y^n); they resolve Problem 1.6 from Vukusic–Ziegler, giving Theorem 1.7 on 2-multiplicatively dependent triples. The elementary lemmas in Sections 2 and 4 are mostly sound and carefully referenced. This is a serious contribution to an established program, not a throwaway.

The soft spots beyond the theorem statement: the computational checks in Propositions 3.1/3.2 are described as 'a quick computation using SageMath' without showing the log or the verification data; Proposition 6.3 depends on the Factoring Database for 102 primes. That's a reproducibility concern, not evidence of fraud, and the code links help. But it means the paper is not fully self-contained.

My take: the error in Theorem 1.3 is specific and likely fixable—the exceptional set needs to be corrected, and the proof's small-case handling needs to be reconciled with the statement. The rest of the paper may well be correct. But as it stands, the main result is false, and no editor should accept it in this form.

A serious referee should see this paper: the methods are valuable, the flaw is concrete and repairable, and the surrounding results deserve scrutiny. After correction, it would be citable. I wouldn't cite the current version.

Referee Report

3 major / 3 minor

Summary. The paper studies triples of integers (a,b,c) with the property that (a,b,c), (a+1,b+1,c+1), and (a+2,b+2,c+2) are each multiplicatively dependent. It claims a complete classification for a=2 in Theorem 1.3 and a classification of the 2-multiplicatively dependent case in Theorem 1.7. The proofs combine elementary arguments, lower bounds for linear forms in logarithms, LLL reduction, and finite computational checks. The main technical ingredients are solutions of two exponential equations in Section 3, a study of x^2 - 2 = y^n in Section 5, and solutions of the three Diophantine equations from Problem 1.6 in Section 6.

Significance. If correct, Theorem 1.3 would complement the finiteness theorem for a>=3 and answer a natural classification question, while Theorem 1.7 would resolve all cases of Problem 1.6. The paper contains detailed arguments and makes a serious effort to supply code and factorization data. However, the central classification in Theorem 1.3 is false as stated, with valid triples omitted and an invalid exception included. This materially undermines the paper's main claim, although the secondary results may still be salvageable after a substantial correction.

major comments (3)

[Theorem 1.3 (Section 1)] Theorem 1.3 is false as stated. The triple (2,3,8) satisfies 2 < b < c and all three hypotheses: (2,3,8) is dependent via 2^3 = 8, (3,4,9) via 3^2 = 9, and (4,5,10) via 4*5^2 = 10^2. It is not in family (1.1), since b=3, not 8; it is not in family (1.2), since b is not of the form 2^x - 2; and it is not one of the listed exceptions. Likewise (2,6,8) is a valid solution: (2,6,8) via 2^3 = 8, (3,7,9) via 3^2 = 9, and (4,8,10) via 4^3 = 8^2, yet it is omitted from the theorem. Conversely, the listed exception (2,6,16) fails, because (3,7,17) consists of distinct primes and is multiplicatively independent. The classification therefore contains both omissions and a false inclusion.
[Section 4.4, first paragraph] The proof of Theorem 1.3 contains an unjustified reduction. After Lemma 4.6, the text says 'we may assume, without loss of generality, that either b or b+2 is a power of two.' Lemma 4.6 only guarantees that one of b, c, b+2, c+2 is a power of two. In the omitted examples (2,3,8) and (2,6,8), the power of two is c=8, and no argument is given for replacing the ordered pair (b,c) by (c,b) while preserving the hypotheses and the classification. This gap is load-bearing: it is exactly what allows the proof to miss these solutions. The WLOG step is invalid as stated.
[Section 4, opening restatement] The proof's restatement of Theorem 1.3 at the beginning of Section 4 contradicts the theorem statement and is itself incorrect. The restatement adds the small exceptions (2,3,8), (2,4,8), and (2,6,8) to family (1.1), while Theorem 1.3 does not include them. Moreover, (2,4,8) is not a solution: the third triple (4,6,10) has prime-exponent vectors (2,0,0), (1,1,0), and (1,0,1), which are linearly independent over Z, so (4,6,10) is multiplicatively independent. Thus even the internal statement used in the proof is not correct.

minor comments (3)

[Section 3, proof of Propositions 3.1 and 3.2] The final computational step is described only as 'A quick computation using SageMath reveals...' with no table of the bounds U(x) or the exceptional candidates. The linked CoCalc page is useful, but the journal version should include the script and output or a compact table of the reduction data to make the finite check reproducible without external links.
[Section 6, proof of Proposition 6.3] The reliance on the Factoring Database [18] is documented with an access date, but for a complete proof the paper should state which factorization data were used and how the entries were verified, or include the relevant factorizations as supplementary material.
[Throughout] There are several typographical and formatting issues, including stray inserted spaces in the title and the phrase 'my assume' in Section 4.4; these should be corrected in any revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the classifications are derived from independent Diophantine lemmas and finite computational checks, not from their own conclusions.

full rationale

No circular reasoning is evident. The paper's central claims are derived from classical external results (Mihailescu, Nagell, Bennett, Ljunggren, Størmer, Zsigmondy), from explicit lower bounds for linear forms in logarithms (Matveev, Laurent), and from finite computational eliminations (SageMath LLL reduction and the Factoring Database). The self-citations to [20] (Lemmas 2.1, 2.2 and Theorem 1.2) are used as external lemmas about consecutive multiplicatively dependent pairs and finiteness for a>=3; they are independently published results and do not assume Theorem 1.3 or Theorem 1.7, so they are not circularly load-bearing. The proof of Propositions 3.1 and 3.2 reduces equation (1.3) to a finite range (3 <= x <= 296) and a finite bound U(x) < 449, then checks candidates; this is an algorithmic elimination, not a fitting of the classification to the answer. Proposition 6.3's check of divisors of 2^{t-1}-1 using the Factoring Database is likewise an external finite verification. One non-circular anomaly is worth flagging separately: the Section 4 restatement of Theorem 1.3 includes exceptional triples (2,3,8), (2,4,8), and (2,6,8), while the introduction's Theorem 1.3 statement does not list them; this is an internal consistency or correctness issue, not a derivation that reduces to its inputs.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper relies on a battery of published theorems from Diophantine analysis, all imported as external results. It introduces no ad hoc postulates or fitted parameters. The key new ingredients are the proof of equations (1.3) and the exponent bound for x^2-2=y^n, both derived from the listed standard tools.

assumptions (9)

standard math Mihailescu's theorem (Theorem 2.3): the only consecutive perfect powers are 8 and 9.
Used to classify solutions to a^x - b^y = 1 throughout the paper, e.g., in Lemma 3.3.
standard math Nagell's theorem (Theorem 2.4): the equation x^n - y^2 = 2 has exactly one solution (3,5,3).
Used in Sections 4 and 6 to solve specific exponential equations.
standard math Bennett's theorem (Theorem 2.5): for fixed a,b >= 2, the equation a^x - b^y = 2 has at most one solution.
Used in the proof of Proposition 6.2 to rule out two solutions to a single equation.
standard math Matveev's lower bound for linear forms in logarithms (Theorem 2.18).
Provides the initial upper bound on variables in the proof of Propositions 3.1 and 3.2.
standard math Laurent's lower bound for linear forms in two logarithms (Theorem 2.20 and Corollary 2.21).
Used to derive the bound n < 1237 for x^2 - 2 = y^n in Proposition 5.1 and in the proof of Propositions 3.1/3.2.
standard math Ljunggren's theorem (Lemma 2.6): the only solution to x^2 + 1 = 2 y^n with n >= 3 is (x,y,n)=(239,13,4).
Used in Case 3.1 of the proof of Lemma 4.6.
standard math Zsigmondy's theorem (Lemma 2.17): primitive prime divisors exist for a^n - b^n and a^n + b^n with stated exceptions.
Used repeatedly to bound exponents in Section 4 and in the proof of Proposition 6.1.
standard math Chen's results on x^2 - 2 = y^p (Propositions 5.3 and 5.4): congruence conditions and y > 10^102.
Critical for the lower bound on y in Proposition 5.1 and for the contradiction in Proposition 6.3.
standard math Vukusic-Ziegler Lemmas 2.1 and 2.2: classification of consecutive and two-shifted multiplicatively dependent pairs.
Used as building blocks in Section 4 and Section 7 to constrain the structure of dependent triples.

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Pith. "Pith review of More on consecutive multiplicatively dependent triples of integers." pith.science (2026). https://pith.science/paper/YRKQHTIR

@misc{pith2026241112009,
  author       = {Pith},
  title        = {Pith review of: More on consecutive multiplicatively dependent triples of integers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YRKQHTIR}},
  note         = {Machine review of arXiv:2411.12009}
}

abstract

In this paper, we extend recent work of the third author and Ziegler on triples of integers $(a,b,c)$, with the property that each of $(a,b,c)$, $(a+1,b+1,c+1)$ and $(a+2,b+2,c+2)$ is multiplicatively dependent, completely classifying such triples in case $a=2$. Our techniques include a variety of elementary arguments together with more involved machinery from Diophantine approximation.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On the Lebesgue-Nagell equation $x^2-2 = y^p$
math.NT 2025-07 conditional novelty 7.0 of 10

For the equation x^2 - 2 = y^p, the authors prove the only solutions for p>911 are y = -1, and any nontrivial solution has y > 10^1000.

Reference graph

Works this paper leans on

23 extracted references · 17 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

M. A. Bennett. On some exponential equations of S. S. Pill ai. Canad. J. Math. , 53(5):897–922, 2001. doi:10.4153/CJM-2001-036-6

work page doi:10.4153/cjm-2001-036-6 2001

[2] [2]

M. A. Bennett. Products of consecutive integers. Bull. London Math. Soc. , 36(5):683–694, 2004. doi:10.1112/S0024609304003480

work page doi:10.1112/s0024609304003480 2004

[3] [3]

M. A. Bennett and C. M. Skinner. Ternary Diophantine equa tions via Galois representations and mod- ular forms. Canad. J. Math. , 56(1):23–54, 2004. doi:10.4153/CJM-2004-002-2

doi:10.4153/cjm-2004-002-2 2004

[4] [4]

Bugeaud, M

Y. Bugeaud, M. Mignotte, and S. Siksek. Classical and mod ular approaches to exponential Diophantine equations. II. The Lebesgue-Nagell equation. Compos. Math. , 142(1):31–62, 2006. doi:10.1112/S0010437X05001739

work page doi:10.1112/s0010437x05001739 2006

[5] [5]

I. Chen. On the equations a2 − 2b6 = cp and a2 − 2 = cp. LMS J. Comput. Math. , 15:158–171, 2012. doi:10.1112/S146115701200006X

doi:10.1112/s146115701200006x 2012

[6] [6]

H. Cohen. Number theory. Vol. I. Tools and Diophantine equations , volume 239 of Graduate Texts in Mathematics. Springer, New York, 2007. doi:10.1007/978-0-387-49923-9

work page doi:10.1007/978-0-387-49923-9 2007

[7] [7]

H. Cohen. Number theory. Vol. II. Analytic and modern tools , volume 240 of Graduate Texts in Math- ematics. Springer, New York, 2007. doi:10.1007/978-0-387-49894-2

doi:10.1007/978-0-387-49894-2 2007

[8] [8]

J. H. E. Cohn. The Diophantine equations x3 = N y2± 1. Quart. J. Math. Oxford Ser. (2) , 42(165):27–30,

work page