# The Erdős-Moser equation 1k+2k+...+(m-1)k=mk revisited using continued fractions

@article{Gallot2011TheEE,
title={The Erdős-Moser equation 1k+2k+...+(m-1)k=mk revisited using continued fractions},
author={Yves Gallot and Pieter Moree and Wadim Zudilin},
journal={Math. Comput.},
year={2011},
volume={80},
pages={1221-1237}
}
• Published 8 July 2009
• Mathematics, Computer Science
• Math. Comput.
If the equation of the title has an integer solution with $k\ge2$, then $m>10^{9.3\cdot10^6}$. This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark $m>10^{10^7}$. Here we achieve $m>10^{10^9}$ by showing that $2k/(2m-3)$ is a convergent of $\log2$ and making an extensive continued fraction digits calculation of $(\log2)/N$, with $N$ an appropriate integer. This method is very different from that of Moser…

## Tables from this paper

On the unsolvability of certain equations of Erdős–Moser type
Let $S_k(m):=\sum_{j=1}^{m-1}j^k$ denote a power sum. In 2011, Kellner proposed the conjecture that for $m>3$ the ratio $S_k(m+1)/S_k(m)$ is never an integer, or, equivalently, that for any positive
Moser's mathemagical work on the equation 1^k+2^k+...+(m-1)^k=m^k
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
A congruence modulo $n^3$ involving two consecutive sums of powers and its applications
For various positive integers $k$, the sums of $k$th powers of the first $n$ positive integers, $S_k(n+1)=1^k+2^k+...+n^k$, have got to be some of the most popular sums in all of mathematics. In this
Primary Pseudoperfect Numbers, Arithmetic Progressions, and the Erdős-Moser Equation
• Mathematics
Am. Math. Mon.
• 2017
It is shown that K is congruent to 6 modulo 36 if 6 divides K, and a remarkable 7-term arithmetic progression of residues modulo 288 in the sequence of known PPNs is uncovered.
Forbidden Integer Ratios of Consecutive Power Sums
• Mathematics
• 2016
Let S k (m): = 1 k + 2 k + ⋯ + (m − 1) k denote a power sum. In 2011 Bernd Kellner formulated the conjecture that for m ≥ 4 the ratio S k (m + 1)∕S k (m) of two consecutive power sums is never an
On a Congruence Modulo n 3 Involving Two Consecutive Sums of Powers
• Mathematics
• 2014
For various positive integers k, the sums of kth powers of the first n positive integers, Sk(n) := 1 k +2 k +···+n k , are some of the most popular sums in all of mathematics. In this note we prove a
A Top Hat for Moser's Four Mathemagical Rabbits
It is shown here that Moser's result can be derived from a von Staudt-Clausen type theorem and the mathematical arguments used in the proofs were already available during the lifetime of Lagrange.
Neverending Fractions: An Introduction to Continued Fractions
• Mathematics
• 2014
Preface 1. Some preliminaries from number theory 2. Continued fractions, as they are 3. Metric theory of continued fractions 4. Quadratic irrationals through a magnifier 5. Hyperelliptic curves and
List of Publications of
According to multivariate regression analysis, systolic and diastolic blood pressure was significantly associated with weight, BMI and waist circumference, and support the hypothesis that BMI is a significant predictor of blood pressure in adolescent age groups.

## References

SHOWING 1-10 OF 44 REFERENCES
Diophantine equations of Erdös-Moser type
• P. Moree
• Mathematics
Bulletin of the Australian Mathematical Society
• 1996
Using an old result of Von Staudt on sums of consecutive integer powers, we shall show by an elementary method that the Diophantine equation 1k + 2k + … + (x − l)k = axk has no solutions (a, x, k)
A Comparative Study of Algorithms for Computing Continued Fractions of Algebraic Numbers
• Mathematics, Computer Science
ANTS
• 1996
The obvious way to compute the continued fraction of a real number α > 1 is to compute a very accurate numerical approximation of α, and then to iterate the well-known truncate-and-invert step which
On Artin's conjecture.
The problem of determining the prime numbers p for which a given number a is a primitive root, modulo JP, is mentioned, for the partieular case a — 10, by Gauss in the section of the Disquisitiones
On Schönhage's algorithm and subquadratic integer gcd computation
A new subquadratic left-to-right GCD algorithm, inspired by Schonhage's algorithm for reduction of binary quadratic forms, is described, which runs slightly faster than earlier algorithms, and is much simpler to implement.
On the equation sum p/N (1/p + 1/N ) = 1, pseudoperfect numbers, and perfectly weighted graphs
• Mathematics
Math. Comput.
• 2000
All solutions to the equation Σ p|N 1/p + 1/N = 1 with at most eight primes are presented, the bound on the nonsolvability of the Erdos-Moser equation is improved, and computational search techniques used to generate examples of perfectly weighted graphs are discussed.
A Note on The Metrical Theory of Continued Fractions
• Mathematics
Am. Math. Mon.
• 2000
The purpose of this note is to prove Davison's conjecture that almost all real a have infinitely many even numbered convergents to their continued fraction expansion with even numerator.
On a theorem of Carlitz
Abstract. Carlitz proved that, for any prime power q > 2, the group of all permutations of 𝔽q is generated by the permutations induced by degree-one polynomials and xq-2. His proof relies on a
Znam's Problem
• Mathematics
• 2002
Fractional expressions of this sort occurred naturally within the Egyptian system of arithmetic. The mathematician-scribes of dynastic Egypt denoted rational numbers by strings of unit fractions
A course in computational algebraic number theory
• H. Cohen
• Computer Science, Mathematics
Graduate texts in mathematics
• 1993
The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.
Unsolved Problems in Number Theory
This monograph contains discussions of hundreds of open questions, organized into 185 different topics. They represent aspects of number theory and are organized into six categories: prime numbers,