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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.

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Divisibility properties of integers $x,\ k$ satisfying $1^ k+\cdots +(x-1)^ k=x^ k$
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by P. Moree, H. J. J. te Riele and J. Urbanowicz PDF
Math. Comp. 63 (1994), 799-815 Request permission


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 $\leq 200$.
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Math. Comp. 63 (1994), 799-815
  • MSC: Primary 11D41; Secondary 11B68, 11Y50
  • DOI:
  • MathSciNet review: 1257577