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Concept
Singly and doubly even
Summary
In mathematics an even integer, that is, a number that is divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not. The former names are traditional ones, derived from ancient Greek mathematics; the latter have become common in recent decades.
These names reflect a basic concept in number theory, the 2-order of an integer: how many times the integer can be divided by 2. This is equivalent to the multiplicity of 2 in the prime factorization.
A singly even number can be divided by 2 only once; it is even but its quotient by 2 is odd.
A doubly even number is an integer that is divisible more than once by 2; it is even and its quotient by 2 is also even.
The separate consideration of oddly and evenly even numbers is useful in many parts of mathematics, especially in number theory, combinatorics, coding theory (see even codes), among others.
The ancient Greek terms "even-times-even" (ἀρτιάκις ἄρτιος) and "even-times-odd" (ἀρτιάκις περισσός or ἀρτιοπέριττος) were given various inequivalent definitions by Euclid and later writers such as Nicomachus. Today, there is a standard development of the concepts. The 2-order or 2-adic order is simply a special case of the p-adic order at a general prime number p; see p-adic number for more on this broad area of mathematics. Many of the following definitions generalize directly to other primes.
For an integer n, the 2-order of n (also called valuation) is the largest natural number ν such that 2ν divides n. This definition applies to positive and negative numbers n, although some authors restrict it to positive n; and one may define the 2-order of 0 to be infinity (see also parity of zero). The 2-order of n is written ν2(n) or ord2(n). It is not to be confused with the multiplicative order modulo 2.
The 2-order provides a unified description of various classes of integers defined by evenness:
Odd numbers are those with ν2(n) = 0, i.e., integers of the form 2m + 1.
Even numbers are those with ν2(n) > 0, i.e.
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