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Mathematics of Computation

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Divisibility properties of integers $x,\ k$ satisfying $1^ k+\cdots +(x-1)^ k=x^ k$


Authors: P. Moree, H. J. J. te Riele and J. Urbanowicz
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
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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 $\leq 200$.


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Keywords: Sums of powers, regular and irregular primes, Bernoulli numbers, congruences, Diophantine equations
Article copyright: © Copyright 1994 American Mathematical Society