Corollary. If is a squarefree positive integer with a given number of prime factors, then the total count of its divisors is a power of two, specifically, two raised to the number of prime factors. With being the Möbius function, being the number of distinct prime factors function, and the divisor function, we can write that if then .
Since it is stipulated that is squarefree, the number of (nondistinct) prime factors function (http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction) can be used instead of .
Note that for the sake of simplicity in this particular application, 1 is not considered a prime number.
has prime factors, which here are labelled (they don’t necessarily have to be sorted, but it doesn’t hurt). This accounts for the prime divisors of , therefore must be at least . The semiprime divisors of are Basically, the number of ways to choose two prime factors from among choices, which is the binomial coefficient
Likewise, the first determination, though quite obvious, was
The binomial coefficients can be used to determine the count of divisors with 3 prime factors, 4 prime factors, etc., all the way up to factors, which is again obvious:
This should be familiar, as these values can be looked up in Pascal’s triangle.
So far we’ve only looked at the divisors of which are the product of one or more prime numbers. But every positive integer has 1 as a divisor, and since 1 has zero prime factors, we can also use Pascal’s triangle to count 1 as a divisor thus: the number of ways to choose zero prime factors from among choices, and that’s the 1 in the leftmost column of row in Pascal’s triangle. So to count the number of divisors of a squarefree number with prime factors, it is a simple matter of adding up row of Pascal’s triangle:
exactly as the theorem stated. ∎
The consequences of the theorem are that the number 1 has one divisor: 1; a prime has two divisors: ; a semiprime has four divisors: ; a sphenic number has eight divisors: ; a number with four prime factors has sixteen divisors; and so on and so forth.
|Date of creation||2013-03-22 18:18:41|
|Last modified on||2013-03-22 18:18:41|
|Last modified by||1and2and4 (20899)|