Centres of mass, Steiner's theorem, and equations of Erdős-Moser type

Abstract

We study the Erdős--Moser equation $1^k+2^k+\dots+(m-1)^k=m^k$ and its analogue in arithmetic progressions, $\sum_{i<m,\ i\equiv b \pmod q} i^k=m^k$, by means of centres of mass, through what we call the half-mass principle. For the progression equation we show that a solution with exponent $k$ has at most $\lfloor k/\log 2+3/2\rfloor$ terms for all $k<10^{55}$, which is best possible for real solutions and replaces an exponential bound; that for $k=3$ solutions with exactly $n$ terms exist if and only if $n\in\{3,4,5\}$, and then infinitely many; and that there are no solutions at all for $k=4$, more generally for every $k=2^s\ge 4$, and for every even $k$ with $4\le k\le 24$. These results rest on a uniform estimate for the Erdős--Moser equation: the real exponent $κ_m$ for which $1^κ+\dots+(m-1)^κ=m^κ$ satisfies $κ_m=\log 2\,(m-3/2-γ_m/m)$ with $0<γ_m<1/(8\log 2)$ for every $m\ge 3$. The method also explains the constants: $3/2$ is a Steiner quotient $(ρ^2+d^2)/2d$, and $25/12-3\log 2=1/12-f''(\log 2)/2$ with $f(λ)=λ/(e^λ-1)$, which explains why this number is so small.