doubly even number

A doubly even number is an even numberMathworldPlanetmath divisible by 4 and sometimes greater powers of two. If n is a doubly even number, it satisfies the congruenceMathworldPlanetmath n0mod4. 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 valueMathworldPlanetmathPlanetmathPlanetmath 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 ( of a doubly even number n with p=2 is always 14 or less.

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


with pi being the ith prime numberMathworldPlanetmath, a1>1, all other other ai may have any nonnegative integer value.

If n is doubly even, then the value of τ(n) (the divisor functionDlmfDlmfMathworldPlanetmath) is even except when all the nonzero ai in the factorization are greater than 1.

Whereas (-1)n=1 whether n is singly or doubly even, with the imaginary unitMathworldPlanetmath i it is the case that in=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