[Paper Review] Moser's mathemagical work on the equation 1^k+2^k+...+(m-1)^k=m^k

Pieter Moree|arXiv (Cornell University)|Nov 12, 2010

Analytic Number Theory Research18 references3 citations

TL;DR

This paper revisits Leo Moser's 1953 result that the Diophantine equation $1^k + 2^k + \cdots + (m-1)^k = m^k$ has no solutions for $k \geq 2$ with $m > 10^{10^6}$, offering a simplified, modernized proof using a von Staudt-Clausen-type theorem and divisibility properties of numbers in the sequence $\{2^{2e+1}+1\}$. It strengthens Moser's bound to $m > 1.485 \times 10^{9,321,155}$ and proves that any solution must satisfy stringent arithmetic conditions on $m$ and $k$, including $k$ even, $m \equiv 3 \pmod{8}$, and $m-1$, $(m+1)/2$, $2m-1$, $2m+1$ being square-free with $p-1 \mid k$ for all prime divisors $p$. The paper also fully solves a class of Erdős-Moser-type equations $aS_k(m) = m^k$ for infinitely many $a$, showing $\mathcal{A}$ is empty under certain conditions.

ABSTRACT

If the equation of the title has an integer solution with k>=2, then m>10^{10^6}. Leo Moser showed this in 1953 by amazingly elementary methods. With the hindsight of more than 50 years his proof can be somewhat simplified. We give a further proof showing that Moser's result can be derived from a von Staudt-Clausen type theorem. Based on more recent developments concerning this equation, we derive a new result using the divisibility properties of numbers in the sequence 2^{2e+1}+1, e=0,1,2,..... In the final section we show that certain Erdos-Moser type equations arising in a recent paper of Kellner can be solved completely.

Motivation & Objective

• To re-derive and simplify Leo Moser's 1953 proof that $1^k + \cdots + (m-1)^k = m^k$ has no solutions for $k \geq 2$ with $m > 10^{10^6}$ using modern number-theoretic tools.
• To improve Moser's bound on $m$ by proving $m > 1.485 \times 10^{9,321,155}$ using refined divisibility and congruence conditions.
• To establish that any solution must satisfy strong arithmetic constraints: $k$ even, $m \equiv 3 \pmod{8}$, $m \equiv \pm1 \pmod{3}$, and several expressions involving $m$ must be square-free.
• To show that certain Erdős-Moser-type equations $aS_k(m) = m^k$ can be solved completely for infinitely many integers $a$, particularly when $a$ satisfies specific congruence and divisibility conditions.

Proposed method

• Reformulating Moser's original proof using a variant of the von Staudt-Clausen theorem to derive bounds on $m$ via $p$-adic valuation and congruence analysis.
• Applying a binomial identity due to Pascal to reprove Lemma 1, which states that $S_k(p) \equiv -1 \pmod{p}$ if $p-1 \mid k$, and $0$ otherwise.
• Using properties of Bernoulli numbers and modular arithmetic to analyze the equation modulo prime powers and derive constraints on $m$ and $k$.
• Analyzing the divisibility structure of numbers of the form $2^{2e+1} + 1$ to derive a new lower bound of $m \geq 10^{10^{16}}$ under the condition that $m+2$ is composed only of primes $p \equiv 5,7 \pmod{8}$.
• Proving that if a prime $q$ divides $a$ with $q^2 \nmid a$ and $q-1 \mid k$, then $aS_k(m) \neq m^k$, thus excluding such $a$ from the solution set.
• Combining analytic techniques, including continued fraction approximations of $\log(1+1/a)$, to explore the feasibility of breaking the $10^{10^7}$ barrier in bounding $m$.

Experimental results

Research questions

• RQ1 Can Moser's original proof of the nonexistence of solutions to $1^k + \cdots + (m-1)^k = m^k$ for $k \geq 2$ be simplified and modernized using contemporary number-theoretic tools?
• RQ2 What is the sharpest known lower bound on $m$ for any solution to the equation, and can it be improved using divisibility properties of $2^{2e+1}+1$?
• RQ3 Under what conditions can the generalized Erdős-Moser equation $aS_k(m) = m^k$ be solved completely for fixed $a$?
• RQ4 What arithmetic constraints must $m$ and $k$ satisfy if a solution exists, particularly regarding squarefreeness and congruence conditions?
• RQ5 Can the method of continued fractions, as used in the computation of $\log 2$, be adapted to improve bounds on $m$ in the original equation?

Key findings

• The paper establishes a new lower bound of $m > 1.485 \times 10^{9,321,155}$ for any solution to $1^k + \cdots + (m-1)^k = m^k$ with $k \geq 2$.
• It proves that any solution must have $k$ even, $m \equiv 3 \pmod{8}$, and $m \equiv \pm1 \pmod{3}$, with $m-1$, $(m+1)/2$, $2m-1$, and $2m+1$ all square-free.
• For any prime $p$ dividing one of the square-free expressions in $m$, it is shown that $p-1$ must divide $k$.
• The number $(m^2 - 1)(4m^2 - 1)/12$ is proven to be square-free and to have at least $4,990,906$ distinct prime factors.
• Under the condition that $m+2$ is composed only of primes $p \equiv 5,7 \pmod{8}$, the bound on $m$ is strengthened to $m \geq 10^{10^{16}}$.
• The paper fully solves the generalized Erdős-Moser equation $aS_k(m) = m^k$ for infinitely many $a$, showing that $a \notin \mathcal{A}$ if $q \mid a$, $q^2 \nmid a$, and $q-1 \mid k$, thus excluding such $a$ from having solutions.