An Analytical Exploration of the Erd\"os-Moser Equation $ \sum_{i=1}^{m-1} i^k = m^k $ Using Approximation Methods

Guillaume Lambard

REVIEW 4 major objections 4 minor 8 references

An Analytical Exploration of the Erd\"os-Moser Equation $ \sum_{i=1}^{m-1} i^k = m^k $ Using Approximation Methods

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper argues that an Euler–MacLaurin approximation of the power sum, combined with the rational root theorem and sign checks at every candidate integer, rules out all solutions of the Erdős–Moser equation with $k \ge 2$, leaving…

desk verdict A self-aware heuristic on the Erdős-Moser equation whose own Section 6.1 concedes the method cannot certify Diophantine non-existence, yet whose strongest sections assert the conjecture anyway. read the letter →

arxiv 2411.13146 v3 pith:ECEB6AFV submitted 2024-11-20 math.NT math.CO

Guillaume Lambard This is my paper · ORCID

classification math.NTmath.CO MSC 11D4111B68

keywords Erdős–MoserequationsumsofpowersEuler–MacLaurinformularationalroottheoremDiophantineequationsBernoullinumbersintegersolutions

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 takes aim at the Erdős–Moser conjecture, which says that the equation $\sum_{i=1}^{m-1} i^k = m^k$ has no positive integer solutions with $k \ge 2$ and $m \ge 3$, leaving $(k,m)=(1,3)$ as the only solution. The author replaces the discrete power sum $S(m-1,k)$ by a continuous Euler–MacLaurin approximation $S_{\mathbb{R}}(m-1,k)$, builds the approximate polynomial $P_{\mathbb{R}}(m)=S_{\mathbb{R}}(m-1,k)-m^k$, and applies the rational root theorem to enumerate the only possible integer roots. Sign checks at each candidate root lead the author to conclude that no positive integer $m \ge 3$ satisfies the approximate equation for $k \ge 2$, and the paper's strongest statements claim this confirms the conjecture and hence that $(1,3)$ is unique. The paper is explicit, however, that this is not a definitive proof: $P_{\mathbb{R}}(m)$ is a truncated approximation, and its roots need not be roots of the exact polynomial $P(m)=S(m-1,k)-m^k$. A correct proof would settle a problem that has been open since the 1950s and currently has only a towering lower bound on any further solution; the paper positions itself as evidence and an analytical case study rather than a closing argument.

What carries the argument

The load-bearing object is the Euler–MacLaurin truncation $P_{\mathbb{R}}(m)$, an approximate stand-in for the exact difference $P(m)=S(m-1,k)-m^k$. After multiplying by $2(k+1)$, the equation $P_{\mathbb{R}}(m)=0$ becomes the integer-coefficient polynomial $2(m-1)^{k+1}+(k+1)(m-1)^k-2(k+1)m^k+(k-1)=0$, and for odd $k$ the paper divides by $m$ to get $Q_{\mathbb{R}}(m)$. The rational root theorem then converts the search over all integers into a finite check: any integer root must divide the constant term (or, for odd $k$, $(k+1)(k-2)$) with the leading coefficient restricted to $2$. The argument's work is done by sign estimates comparing the positive terms $(m-1)^{k+1}$ and $(m-1)^k$ against the negative term $m^k$ at the surviving candidate points, using limits like $(1+1/(k-2))^k \to e$ to show the sign cannot change.

What would settle it

Compute the exact difference $P(m)=\sum_{i=1}^{m-1} i^k - m^k$ at the candidate values $m_0 \in \{k-1, 2(k-1), k-2, k+1, (k+1)(k-2)\}$ for a range of $k$, using exact arithmetic via Faulhaber's formula. If any $P(m_0)=0$, the conjecture is false and the paper's conclusion collapses. Short of that, if the sign of $P(m_0)$ ever differs from the sign of $P_{\mathbb{R}}(m_0)$ reported in the paper's lemmas, the approximation is demonstrably unreliable at exactly the points the argument depends on, and the proof by approximation fails even if the conjecture remains true.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that a rational-root-theorem analysis of the approximate polynomial $P_{\mathbb{R}}(m) = \frac{(m-1)^{k+1}}{k+1} + \frac{(m-1)^k}{2} - m^k - \frac{1}{k+1} + \frac{1}{2}$ (with denominators cleared) leaves only a small, explicit set of candidate integer roots for $k \ge 2$: for even $k \ge 4$, $m_0 = k-1$ or $2(k-1)$; for odd $k \ge 5$, $m_0 = k-2$; and for odd $k \ge 3$, $m_0 = k+1$ or $(k+1)(k-2)$. Evaluating $P_{\mathbb{R}}$ at these candidates and comparing the dominant terms shows each has a definite nonzero sign, so the approximate polynomial never vanishes at an integer $m \ge 3$. In the full Euler–MacLaurin version, the paper argues that including all Bernoulli correction terms yields integer coefficients whose constant term and leading coefficient grow so fast that the finite set of rational candidates cannot contain an integer root. The paper's conclusion is uniqueness of $(1,3)$; the qualification attached throughout is that these are statements about the approximation unless the omitted correction terms are shown to be harmless at integer points.

Load-bearing premise

The entire chain of candidate roots and sign checks is performed on the truncated polynomial $P_{\mathbb{R}}(m)$, and the paper itself states in Section 6.1 that because $P_{\mathbb{R}}$ comes from an approximation, its roots may not be roots of the exact polynomial $P(m)=S(m-1,k)-m^k$; if the omitted Euler–MacLaurin correction terms move the function across zero at an integer, a genuine solution could be hiding right where the approximation says there is none.

Editorial extensions

If this is right

For $k=1$ the exact equation reduces to $m^2-3m=0$, so $m=3$ is the unique solution, confirming the conjecture in that case.
If the rational-root enumeration and sign checks are correct for the approximate polynomial, then no candidate integer $m \ge 3$ survives for any $k \ge 2$, so the approximate equation has no integer solutions.
The full Euler–MacLaurin version with all Bernoulli corrections is argued to preserve this conclusion: the integer-coefficient polynomial $\tilde{P}(m)$ has candidate sets that shrink relative to the growth of the coefficients, so no new integer root appears.
The paper's strongest reading is that the Erdős–Moser conjecture is true and $(k,m)=(1,3)$ is the only positive integer solution.
Because the analysis is explicitly approximate, the theorem-level consequence the author claims is evidence rather than proof: any rigorous resolution still requires controlling all Euler–MacLaurin correction terms or an exact method.

Reading between the lines

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

A natural test of the paper's logic is to compute the exact value of $P(m_0)$ at the same candidate points using Faulhaber's formula; if the sign of the exact polynomial ever disagrees with the sign of $P_{\mathbb{R}}(m_0)$ at a candidate, the approximation-based exclusion is falsified even though the conjecture may still be true.
The rational root theorem is being applied to a polynomial whose constant term and leading coefficient come from an approximation; an integer root of the exact $P(m)$ would only be guaranteed to appear among the candidates if the approximation were exact, so the candidate list is a property of $P_{\mathbb{R}}$, not of the Diophantine equation.
The existing lower bounds mean any counterexample, if it exists, is astronomically large; the paper's asymptotic sign analysis around $m \approx \frac{3}{2}(k+1)$ places the approximate crossing where $m$ is comparable to $k$, far below that regime, which is why approximation alone cannot touch the hard part of the conjecture.
One could turn the procedure into a certified method by bounding the exact remainder of the Euler–MacLaurin expansion for all integers $m \ge 3$: if the remainder sign is controlled on the gaps between candidates, the same rational-root and sign scheme would give a genuine proof rather than a heuristic.

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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 honest summary of this paper is written by its own author. Section 6.1 concedes that PR(m) is derived from an approximation, so its roots need not be roots of the exact polynomial P(m) = S(m−1,k) − m^k. That concession undercuts the central inference, and everything else follows from it.

What the paper does well: it is clearly organized, the k=1 case is handled exactly, the full Euler-MacLaurin expansion in Section 5 is the right tool for getting an exact polynomial, and the literature engagement is genuine — Moser, Gallot–Moree–Zudilin, Moree, and Baoulina–Moree are all on point and cited correctly. The abstract and conclusion are honest: this is an exploration, not a proof. The ratio-and-sign analysis in Section 4 is mostly careful calculus.

But the structure cannot support the conclusions drawn. First, the rational root theorem applied to the truncated polynomial says nothing about integer roots of the exact equation. The failure is concrete: for k=2, the exact polynomial is (2m^3 − 9m^2 + m)/6, which vanishes at m=0, while the paper's cleared approximate polynomial is 2m^3 − 9m^2 + 2, with constant term 2. The two polynomials have different candidate sets before the analysis even starts. Second, even inside the approximate framework, Lemma 8's reduction to integer candidates is incomplete: for k=7, (k+1)(k−2) = 40 gives integer candidates 4, 10, and 20, none of which appears in Lemma 14's sign analysis. Third, Section 5's extended analysis never tests the exact polynomial's rational candidates. The divisor-growth discussion shows the candidate list is short, not that it contains no root; and the asymptotic analysis shows PR changes sign near (3/2)(k+1), so a real root exists there. The paper then needs to rule out that this real root is an integer, which it never does. Section 5.6.6's 'completing the proof' is unsupported.

The result is an internal contradiction: Section 6.1 disowns the load-bearing premise, yet Section 4.3.7 claims to 'confirm the validity of Conjecture 1' and Section 5.6.6 asserts the conjecture outright. The strong claims are where the disclaimers should have been.

The paper is not circular and not deceptive — it is a self-aware heuristic. But the heuristic adds nothing to the constraints Moser and Gallot–Moree–Zudilin already proved, and the flaws are load-bearing rather than fixable by revision. For a reading group it could be a useful case study in why analytic approximation cannot certify Diophantine non-existence. As a submission, I would desk-reject it: the referee would only confirm what the author already wrote in Section 6.1.

Referee Report

4 major / 4 minor

Summary. The paper investigates the Erdős-Moser equation S(m-1,k)=m^k using the Euler-MacLaurin formula. The authors define an approximate polynomial P_R(m) from the integral and boundary terms of the Euler-MacLaurin expansion, apply the rational root theorem to P_R(m), and analyze the signs of P_R at a short list of candidate integer roots. They conclude, in Section 4.3.7, that no positive integers m>=3 satisfy the equation for k>=2, and in Section 5.6.6 that the only positive integer solution is (k,m)=(1,3). The paper simultaneously acknowledges in Sections 1 and 6.1 that the approximation method does not constitute a definitive proof and that roots of P_R need not correspond to roots of the exact polynomial P(m)=S(m-1,k)-m^k. The body also contains an 'extended' analysis of an exact Euler-MacLaurin polynomial, but that analysis never rules out the integer candidates for the exact polynomial.

Significance. If the central claim were correct, it would resolve a long-standing open problem in number theory. However, the paper explicitly disclaims a proof, and the main argument suffers from a load-bearing gap: the rational-root theorem is applied to an approximate polynomial whose roots, as the authors concede in Section 6.1, need not coincide with roots of the exact Diophantine polynomial. The paper provides no machine-checked proofs, no reproducible code, and no parameter-free derivation that survives scrutiny; the heuristic value is limited because the omitted Euler-MacLaurin terms can be decisive at integer points. The manuscript is best regarded as an exploration of an approximation, not as a proof of non-existence.

major comments (4)

[Section 4.2.5 / Lemma 8] The candidate list of integer rational roots of P_R is incomplete, so the sign analysis in Section 4.3 does not rule out all rational roots even of the approximate polynomial. For even k, the constant term is 2(k-1) and the leading coefficient is 2, so integer m0 with q=1 or q=2 are all divisors of k-1 and twice those divisors. The paper restricts to {k-1, 2(k-1)}, omitting, for example, m=3 when k=10 (where k-1=9 has divisor 3). For odd k=7, a0=(k+1)(k-2)=40, and the integer candidates include 4, 5, 8, 10, 20, 40; the paper analyzes only 5, 8, 40 and omits 4, 10, 20. Thus Lemma 8's list is not the set of all rational roots of P_R, and the subsequent 'non-vanishing at candidates' argument is incomplete.
[Section 2.2.6 / Section 6.1] The central inference from non-vanishing of P_R to non-existence of integer solutions of P(m)=0 is invalid. Section 2.2.6 states that the approximation allows the rational root theorem to be applied, but Section 6.1 explicitly concedes that 'since P_R(m) is derived from an approximation, its roots may not correspond to those of the exact polynomial P(m)'. A concrete illustration is k=2: the exact polynomial P(m) has constant term 0, while the paper's P_R after clearing denominators has constant term 2; hence the rational-root candidate sets of P and P_R already differ at the constant term. The conclusion of Section 4.3.7 ('no positive integers m >= 3 satisfy ...') therefore does not follow from the analysis of P_R.
[Section 5] The 'extended analysis' using the full Euler-MacLaurin expansion does not fill the gap. Section 5.5 only estimates the number of divisors of the leading coefficient and constant term, without deriving the actual rational-root candidate list for the exact integer polynomial and without checking that any of those candidates fails to be a root. Section 5.6 shows that P_R changes sign for real m and then asserts, in Section 5.6.5, that because the number of possible integer roots is 'severely limited' no integer roots exist; this is a non sequitur, since a sign change only guarantees a real root, not an integer root, and no exclusion of the specific integer candidates is carried out.
[Section 4.3.7 / Section 5.6.6 / Section 6.1] The paper is internally inconsistent about its conclusion. Section 4.3.7 and Section 5.6.6 assert the full Erdős-Moser conjecture (only (1,3) is a solution), while Section 6.1 states that the approximation 'can lead to incorrect conclusions about the existence of integer solutions' and that 'its roots may not correspond to those of the exact polynomial'. Because the body's final assertions rely on the disavowed approximation premise, the paper does not provide a coherent proof of the conjecture; it provides at most a heuristic suggestion, which the abstract itself acknowledges.

minor comments (4)

[Throughout] The manuscript contains numerous typographical and OCR artifacts, including 'F or' for 'For', 'Erd¨ os' in running text, and inconsistent mathematical spacing such as 'Pm−1'.
[Section 2.2.3] The discussion of the Euler-MacLaurin remainder is self-contradictory: it says all nonzero correction terms up to order k make the remainder zero, but then says these correction terms are omitted 'for simplicity'. Omitting nonzero terms cannot leave the remainder zero; the text should either include the terms or clearly state that the approximation error is nonzero.
[Section 5.2 / Section 5.3] The same symbol P_R is used for the approximate polynomial in Section 2 and for the exact polynomial S(m-1,k)-m^k in Section 5; this overloads the notation and makes Section 5.6.1 ambiguous about which polynomial is being analyzed.
[Figures 1 and 2] Figure captions describe curves for ranges of k and m, but the figures are not included in the manuscript text; the captions should be completed or the figures should be supplied to make the graphical claims verifiable.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the approximation-based analysis is a correctness gap, not a self-referential derivation; the paper itself concedes the key limitation.

full rationale

Walking the derivation chain, the paper does not fit parameters to data, does not define its target in terms of its inputs, and does not rely on load-bearing self-citations. The approximation PR(m) = SR(m - 1, k) - m^k is an Euler-MacLaurin truncation of P(m) = S(m - 1, k) - m^k, obtained by a standard formula cited to textbooks; it is not constructed from the desired conclusion. The rational root theorem is then applied to PR(m), and candidate integer roots are tested by sign. That argument is not circular, but it is logically insufficient: roots of a truncated approximate polynomial need not be roots of the exact polynomial. The manuscript itself flags this in Section 6.1: "Since PR(m) is derived from an approximation, its roots may not correspond to those of the exact polynomial P(m) = S(m - 1, k) - m^k." This is an honest admission of a non-circular but decisive correctness defect. The only mildly self-referential element is that the paper's own concluding claims in Sections 4.3.7 and 5.6.6 go beyond the approximation it earlier disowns, but that is an unsupported inference, not a circular definition or a fitted-input-as-prediction step. Accordingly, the circularity score is low; the serious problems with the paper are correctness issues, which fall outside the circularity axis.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's conclusions rest on standard tools (Euler-MacLaurin, rational root theorem) plus two assumptions that are not established: that the truncated approximation preserves integer-root behavior, and that growth-rate divisor-count arguments can replace an explicit check of rational-root candidates. The second is especially fragile because the paper's own candidate lists are incomplete.

assumptions (4)

standard math Euler-MacLaurin formula gives a valid proxy for the discrete sum
Used as the foundation in Lemma 1 and Section 5.1.
standard math Rational root theorem constrains integer roots of the approximate polynomial PR(m)
Applied in Sections 4.2.2 to 4.2.5 and 5.5; the theorem is valid for exact polynomials with integer coefficients.
ad hoc to paper Non-vanishing of PR at candidate roots implies non-vanishing of the exact polynomial P
This is the central assumption in Sections 4.3 and 4.4. The paper itself disclaims it in Section 6.1 by noting that roots of PR may not correspond to roots of P.
ad hoc to paper Growth-rate arguments for a0 and an suffice to exclude all integer roots without enumerating them
Section 5.5 and 5.6 conclude non-existence from divisor-count growth without actually testing the finite set of divisors.

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Cite this review

Pith. "Pith review of An Analytical Exploration of the Erd\"os-Moser Equation $ \sum_{i=1}^{m-1} i^k = m^k $ Using Approximation Methods." pith.science (2026). https://pith.science/paper/ECEB6AFV

@misc{pith2026241113146,
  author       = {Pith},
  title        = {Pith review of: An Analytical Exploration of the Erd\"os-Moser Equation $ \sum_i=1^m-1 i^k = m^k $ Using Approximation Methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ECEB6AFV}},
  note         = {Machine review of arXiv:2411.13146}
}

abstract

The Erd\"{o}s-Moser equation $ \sum_{i=1}^{m - 1} i^k = m^k $ is a longstanding challenge in number theory, with the only known integer solution being $ (k,m) = (1,3) $. Here, we investigate whether other solutions might exist by using the Euler-MacLaurin formula to approximate the discrete sum $ S(m-1,k) $ with a continuous function $ S_{\mathbb{R}}(m-1,k) $. We then analyze the resulting approximate polynomial $ P_{\mathbb{R}}(m) = S_{\mathbb{R}}(m-1,k) - m^k $ under the rational root theorem to look for integer roots. Our approximation confirms that for $ k=1 $, the only solution is $ m=3 $, and for $ k \geq 2 $ it suggests there are no further positive integer solutions. However, because Diophantine problems demand exactness, any omission of correction terms in the Euler-MacLaurin formula could mask genuine solutions. Thus, while our method offers valuable insights into the behavior of the Erd\"{o}s-Moser equation and illustrates the analytical challenges involved, it does not constitute a definitive proof. We discuss the implications of these findings and emphasize that fully rigorous approaches, potentially incorporating prime-power constraints, are needed to conclusively resolve the conjecture.

Figures

Figures reproduced from arXiv: 2411.13146 by the authors.

**Figure 1.** Top: Plots of SR(m − 1, k) (continuous line), mk (dashed line), and Pm−1 i=1 i k (dotted line) as a function of k ∈ [2, 102] (color-coded from purple to yellow) and m ∈ [3, 200] (as the variable on the x-axis). Bottom: Plots of SR(m−1, k)−mk (continuous line), SR(m−1, k)−mk +C (dashed line), and Pm−1 i=1 i k −mk (dotted line) as a function of k ∈ [2, 102] (color-coded from purple to yellow) and m ∈ [3, 200] (as the… view at source ↗

**Figure 2.** Plot of PR(m) as a function of k for the various rational roots m0 described in 4.2.5 [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

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Reference graph

Works this paper leans on

8 extracted references · 7 canonical work pages

[1]

Introduction to analytic number theory

Tom M Apostol. Introduction to analytic number theory . Springer Science & Business Media, 2013

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[2]

Forbidden integer ratios of consecutive power sums

Ioulia N Baoulina and Pieter Moree. Forbidden integer ratios of consecutive power sums. From Arithmetic to Zeta-Functions: Number Theory in Memory of Wolfgang Schwarz , pages 1--30, 2016

work page 2016
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The erd o s-moser equation 1 k+ 2 k+ +(m--1) k= m k revisited using continued fractions

Yves Gallot, Pieter Moree, and Wadim Zudilin. The erd o s-moser equation 1 k+ 2 k+ +(m--1) k= m k revisited using continued fractions. Mathematics of computation , pages 1221--1237, 2011

work page 2011
[4]

Maclaurin's second formula and its generalization

Henry W Gould and William Squire. Maclaurin's second formula and its generalization. The American Mathematical Monthly , 70(1):44--52, 1963

work page 1963
[5]

Jeremy D. King. 90.63 integer roots of polynomials. The Mathematical Gazette , 90(519):455–456, 2006

work page 2006
[6]

Theory and application of infinite series

Konrad Knopp. Theory and application of infinite series . Courier Corporation, 1990

work page 1990
[7]

Moser's mathemagical work on the equation 1 k+ 2 k+ +(m-1) k= m k

Pieter Moree. Moser's mathemagical work on the equation 1 k+ 2 k+ +(m-1) k= m k. The Rocky Mountain Journal of Mathematics , pages 1707--1737, 2013

work page 2013
[8]

On the diophantine equation 1n+ 2n+ 3n+ +(m- 1) n= mn

Leo Moser. On the diophantine equation 1n+ 2n+ 3n+ +(m- 1) n= mn. Scripta Math , 19:84--88, 1953

work page 1953

Pith tools

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

[1] [1]

Introduction to analytic number theory

Tom M Apostol. Introduction to analytic number theory . Springer Science & Business Media, 2013

2013

[2] [2]

Forbidden integer ratios of consecutive power sums

Ioulia N Baoulina and Pieter Moree. Forbidden integer ratios of consecutive power sums. From Arithmetic to Zeta-Functions: Number Theory in Memory of Wolfgang Schwarz , pages 1--30, 2016

work page 2016

[3] [3]

The erd o s-moser equation 1 k+ 2 k+ +(m--1) k= m k revisited using continued fractions

Yves Gallot, Pieter Moree, and Wadim Zudilin. The erd o s-moser equation 1 k+ 2 k+ +(m--1) k= m k revisited using continued fractions. Mathematics of computation , pages 1221--1237, 2011

work page 2011

[4] [4]

Maclaurin's second formula and its generalization

Henry W Gould and William Squire. Maclaurin's second formula and its generalization. The American Mathematical Monthly , 70(1):44--52, 1963

work page 1963

[5] [5]

Jeremy D. King. 90.63 integer roots of polynomials. The Mathematical Gazette , 90(519):455–456, 2006

work page 2006

[6] [6]

Theory and application of infinite series

Konrad Knopp. Theory and application of infinite series . Courier Corporation, 1990

work page 1990

[7] [7]

Moser's mathemagical work on the equation 1 k+ 2 k+ +(m-1) k= m k

Pieter Moree. Moser's mathemagical work on the equation 1 k+ 2 k+ +(m-1) k= m k. The Rocky Mountain Journal of Mathematics , pages 1707--1737, 2013

work page 2013

[8] [8]

On the diophantine equation 1n+ 2n+ 3n+ +(m- 1) n= mn

Leo Moser. On the diophantine equation 1n+ 2n+ 3n+ +(m- 1) n= mn. Scripta Math , 19:84--88, 1953

work page 1953