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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 the ancient Greek; 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. ==Definitions== The ancient Greek terms "even-times-even" and "even-times-odd" 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 . *Even numbers are those with ν2(''n'') > 0, i.e., integers of the form . In particular: * *Singly even numbers are those with ν2(''n'') = 1, i.e., integers of the form . * *Doubly even numbers are those with ν2(''n'') > 1, i.e., integers of the form . * * *In this terminology, a doubly even number may or may not be divisible by 8, so there is no particular terminology for "triply even" numbers. One can also extend the 2-order to the rational numbers by defining ν2(''q'') to be the unique integer ν where : and ''a'' and ''b'' are both odd. For example, half-integers have a negative 2-order, namely −1. Finally, by defining the 2-adic norm, : one is well on the way to constructing the 2-adic numbers. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「singly and doubly even」の詳細全文を読む スポンサード リンク
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