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In 1808 Legendre [1] , p. 10, (
) showed that
Theorem. The exact power of a prime
dividing
,
, is
![]() | (1) |
where denotes the integer part function
and the sum is finite containing
terms, where
is the largest power of
dividing
, i.e.
but
.
The proof is simple. We can suppose that . The value
counts the number of multiples of
less or equal to
. Thus there are
positive integers
with
being the exact power of
dividing each. So the power of
dividing
equals
Corollary 1 (Legendre). Let be a prime and
is the base-
expansion of
. Then
![]() | (2) |
Proof. If , then
,
, and the result holds. Let
. If
, then
Then
Adding columnwise we get
Corollary 2 (Čebyšev identity). If then
![]() | (3) |
Corollary 3. The series diverges (p runs over all primes).
Proof. There follows from (2) that and consequently
or that
Then
Since , the result follows.
Corollary 4. If denotes the sums of digits function in base
then
[1] | Legendre, A. M. (1808). Essai sur la theorie des nombres (2ed.). Paris. |
Cite this web-page as:
Štefan Porubský: Prime power dividing a factorial.