the prime power dividing a factorial
In 1808, Legendre showed that the exact power of a prime dividing is
where is the largest power of being .
If then doesn’t divide , and its power is 0, and the sum
empty. So let the prime .
For each , there are numbers between 1 and with being the greatest power of dividing each. So the power of dividing is
But each in the sum appears with factors and , so the above sum equals
where denotes the sums of digits function in base .
If , then and is . So we assume .
Let be the -adic representation of . Then
|Title||the prime power dividing a factorial|
|Date of creation||2013-03-22 13:22:34|
|Last modified on||2013-03-22 13:22:34|
|Owner||Thomas Heye (1234)|
|Last modified by||Thomas Heye (1234)|
|Author||Thomas Heye (1234)|