Centres of mass, Steiner's theorem, and equations of Erdős-Moser type
Abstract
We study the Erdős-Moser equation in arithmetic progressions, ∑ ik=mk, where i runs over the positive integers i<m with i≡ b pmod q. We prove that a solution has at most ⌊ k/log 2+3/2⌋ terms for all k<1055, and at most k/log 2+5/2 terms in general; this linear bound is best possible for real solutions and replaces the previously known exponential bound (k+1)22k+1. For k=3 we obtain a complete classification: solutions with exactly n terms exist if and only if n∈{3,4,5}, and in each case there are infinitely many primitive solutions. For even exponents there are no solutions with three terms, and there are no solutions at all for k=4, for every k=2s≥ 4, and for every even k with 4≤ k≤ 24. The key analytic input is a uniform estimate for the classical equation 1k+…+(m−1)k=mk: the real exponent κm for which 1κ+…+(m−1)κ=mκ satisfies κm=log 2 (m−3/2−γm/m) with 0<γm<1/(8log 2) for every m≥ 3. The proofs rest on a method based on centres of mass, Steiner's parallel-axis theorem and what we call the half-mass principle. The same framework identifies the constants of a general class of balance equations through the moments of a mass distribution: 3/2 is a Steiner quotient, and 25/12−3log 2=1/12−f''(log 2)/2 with f(λ)=λ/(eλ−1), whose Bernoulli expansion explains why this constant is so small.
Related
- Lehto-type growth conditions and a Picard theorem for the gradient of entire solutions to the Euler -- Lagrange equations of variational integrals with linear growth in the plane including a small extension of Moser's Bernstein result
- Erdős-Ko-Rado properties of Steiner 2-designs
- ON THE GENERAL ERDŐS-MOSER EQUATION VIA THE NOTION OF OLLOIDS
- Well--posedness for the continuum Calogero--Moser equations by parabolic regularization
- The Erdős--Sós Theorem
BibTeX
@misc{mehdinezhad2026centres,
title = {{Centres of mass, Steiner's theorem, and equations of Erdős-Moser type}},
author = {Elham Mehdi-Nezhad and Amir M. Rahimi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39315},
eprint = {2610.39315},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39315},
}
RIS
TY - UNPB
TI - Centres of mass, Steiner's theorem, and equations of Erdős-Moser type
AU - Elham Mehdi-Nezhad
AU - Amir M. Rahimi
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39315
UR - https://arxiv.org/abs/2610.39315
AB - We study the Erdős-Moser equation in arithmetic progressions, ∑ i^k=m^k, where i runs over the positive integers i<m with i≡ b pmod q. We prove that a solution has at most ⌊ k/log 2+3/2⌋ terms for all k<10^55, and at most k/log 2+5/2 terms in general; this linear bound is best possible for real solutions and replaces the previously known exponential bound (k+1)2^2k+1. For k=3 we obtain a complete classification: solutions with exactly n terms exist if and only if n∈{3,4,5}, and in each case there are infinitely many primitive solutions. For even exponents there are no solutions with three terms, and there are no solutions at all for k=4, for every k=2^s≥ 4, and for every even k with 4≤ k≤ 24. The key analytic input is a uniform estimate for the classical equation 1^k+…+(m−1)^k=m^k: the real exponent κ_m for which 1^κ+…+(m−1)^κ=m^κ satisfies κ_m=log 2 (m−3/2−γ_m/m) with 0<γ_m<1/(8log 2) for every m≥ 3. The proofs rest on a method based on centres of mass, Steiner's parallel-axis theorem and what we call the half-mass principle. The same framework identifies the constants of a general class of balance equations through the moments of a mass distribution: 3/2 is a Steiner quotient, and 25/12−3log 2=1/12−f''(log 2)/2 with f(λ)=λ/(e^λ−1), whose Bernoulli expansion explains why this constant is so small.
ER -