|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
parity of  function
|
(Feature)
|
|
|
If the prime factor decomposition of a positive integer $n$ is
 |
(1) |
then all positive divisors of $n$ are of the form $$ p_1^{\nu_1}p_2^{\nu_2}\cdots p_r^{\nu_r} \quad\mbox{where}\quad 0 \le \nu_i \le \alpha_i \quad(i = 1,\,2,\,\ldots,\,r). $$ Thus the total number of the divisors is
 |
(2) |
From this we see that in order to $\tau(n)$ be an odd number, every sum $\alpha_i\!+\!1$ shall be odd, i.e. every exponent $\alpha_i$ in (1) must be even. It means that $n$ has an even number of each of its prime divisors
$p_i$ ; so $n$ is a square of an integer, a perfect square.
Consequently, the number of all positive divisors of an integer is always even, except if the integer is a perfect square.
Examples. 15 has four positive divisors 1, 3, 5, 15 and the square number 16 five divisors
1, 2, 4, 8, 16.
|
"parity of  function" is owned by pahio.
|
|
(view preamble | get metadata)
See Also:  function
| Keywords: |
square of an integer |
This object's parent.
|
|
Cross-references: perfect square, square, prime divisors, even number, even, exponent, odd, sum, odd number, number, divisors, integer, positive, decomposition, prime factor
There is 1 reference to this entry.
This is version 3 of parity of function, born on 2009-05-14, modified 2009-05-15.
Object id is 11781, canonical name is ParityOfTauFunction.
Accessed 337 times total.
Classification:
| AMS MSC: | 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
