doubly even number

A doubly even number is an even number divisible by 4 and sometimes greater powers of two. If $n$ is a doubly even number, it satisfies the congruence $n\equiv 0\mod 4$ . The first few positive doubly even numbers are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, listed in A008586 of Sloane’s OEIS.

In the binary representation of a positive doubly even number, the two least significant bits are always both 0. Thus it takes at least a 2-bit right shift to change the parity of a doubly even number to odd. These properties obviously also hold true when representing negative numbers in binary by prefixing the absolute value with a minus sign. As it turns out, all this also holds true in two’s complement. Independently of binary representation, we can say that the $p$ -adic valuation (http://planetmath.org/PAdicValuation) of a doubly even number $n$ with $p=2$ is always $\frac{1}{4}$ or less.

All doubly even numbers are composite. In representing a doubly even number $n$ as

$\prod_{i=1}^{\pi(n)}{p_{i}}^{a_{i}},$

with $p_{i}$ being the $i$ th prime number, $a_{1}>1$ , all other other $a_{i}$ may have any nonnegative integer value.

If $n$ is doubly even, then the value of $\tau(n)$ (the divisor function) is even except when all the nonzero $a_{i}$ in the factorization are greater than 1.

Whereas $(-1)^{n}=1$ whether $n$ is singly or doubly even, with the imaginary unit $i$ it is the case that $i^{n}=1$ only when $n$ is doubly even.

Title	doubly even number
Canonical name	DoublyEvenNumber
Date of creation	2013-03-22 18:09:38
Last modified on	2013-03-22 18:09:38
Owner	1and2and4 (20899)
Last modified by	1and2and4 (20899)
Numerical id	5
Author	1and2and4 (20899)
Entry type	Definition
Classification	msc 11A63
Classification	msc 11A51
Related topic	SinglyEvenNumber
Related topic	FactorsWithMinusSign