Floor Function is Replicative

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Theorem

The floor function is a replicative function in the sense that:

$\ds \forall n \in \Z_{> 0}: \sum_{k \mathop = 0}^{n - 1} \floor {x + \frac k n} = \floor {n x}$


Proof

\(\ds \sum_{k \mathop = 0}^{n - 1} \floor {x + \dfrac k n}\) \(=\) \(\ds \floor {x n + \floor {x + 1} \paren {\ceiling n - n} }\) Summation over $k$ of $\floor {x + \dfrac k y}$, setting $y = n$
\(\ds \) \(=\) \(\ds \floor {x n + \floor {x + 1} \paren {n - n} }\) Real Number is Integer iff equals Ceiling
\(\ds \) \(=\) \(\ds \floor {n x + \floor {x + 1} \times 0}\)
\(\ds \) \(=\) \(\ds \floor {n x}\)

$\blacksquare$


Historical Note

The fact that the Floor Function is Replicative was discovered by Charles Hermite, who published this result in $1884$.


Sources

  • 1884: Charles HermiteSur quelques conséquences arithmétiques des Formules de la théorie des fonctions elliptiques (Acta Math. Vol. 5: pp. 297 – 330)