2

Arithmetical functions

**2.1 The function [***x***]**

For any real *x*, one signifies by [*x*] the largest integer ≤ *x*, that is, the unique integer such that *x* − 1 < [*x*] ≤ *x*. The function is called ‘the integral part of *x*’. It is readily verified that [*x* + *y*] ≥ [*x*] + [*y*] and that, for any positive integer *n*, [*x* + *n*] = [*x*] + *n* and [*x*/*n*] = [[*x*]/*n*]. The difference *x* − [*x*] is called ‘the fractional part of *x*’; it is written {*x*} and satisfies 0 ≤ {*x*} < 1.

Let now *p* be a prime. The largest integer *l* such that *p*^{l} divides *n*! can be neatly expressed in terms of the above function. In fact, on noting that [*n*/*p*] of the numbers 1, 2, …, *n* are divisible by *p*, that [*n*/*p*^{2}] are divisible by *p*^{2}, and so on, we obtain

It follows easily that *l* ≤ [*n*/(*p* − 1)]; for the latter ...

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