Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Divisibility properties of integers $x,\ k$ satisfying $1^ k+\cdots +(x-1)^ k=x^ k$
HTML articles powered by AMS MathViewer

by P. Moree, H. J. J. te Riele and J. Urbanowicz PDF
Math. Comp. 63 (1994), 799-815 Request permission

Abstract:

Based on congruences $\bmod \;p$ and on properties of Bernoulli polynomials and Bernoulli numbers, several conditions are derived for x, $x,k \geq 2$ to satisfy the Diophantine equation ${1^k} + {2^k} + \cdots + {(x - 1)^k} = {x^k}$. It is proved that ${\text {ord}_2}(x - 3) = {\text {ord}_2}k + 3$ and that x cannot be divisible by any regular prime. Furthermore, by using the results of experiments with the above conditions on an SGI workstation it is proved that x cannot be divisible by any irregular prime $< 10000$ and that k is divisible by the least common multiple of all the integers $\leq 200$.
References
Similar Articles
Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Math. Comp. 63 (1994), 799-815
  • MSC: Primary 11D41; Secondary 11B68, 11Y50
  • DOI: https://doi.org/10.1090/S0025-5718-1994-1257577-1
  • MathSciNet review: 1257577