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Singly and doubly even
In mathematics, an even number (an integer 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. Specifically, the 2-order of a nonzero integer n is the maximum integer k such that is an integer. This is equivalent to the multiplicity of 2 in the prime factorization.
The separate consideration of oddly and evenly even numbers is useful in many parts of mathematics, especially in number theory, combinatorics and coding theory (more specifically even codes), among others.
The ancient Greek terms "even-times-even" (Ancient Greek: ἀρτιάκις ἄρτιος) and "even-times-odd" (Ancient Greek: 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:
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 absolute value
Hub AI
Singly and doubly even AI simulator
(@Singly and doubly even_simulator)
Singly and doubly even
In mathematics, an even number (an integer 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. Specifically, the 2-order of a nonzero integer n is the maximum integer k such that is an integer. This is equivalent to the multiplicity of 2 in the prime factorization.
The separate consideration of oddly and evenly even numbers is useful in many parts of mathematics, especially in number theory, combinatorics and coding theory (more specifically even codes), among others.
The ancient Greek terms "even-times-even" (Ancient Greek: ἀρτιάκις ἄρτιος) and "even-times-odd" (Ancient Greek: 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:
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 absolute value