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 n 2 k {\displaystyle {\frac {n}{2^{k}}}} is an integer. 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 and coding theory (more specifically even codes), among others.
Definitions 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:
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., integers of the form 2m. In particular: Singly even numbers are those with ν2(n) = 1, i.e., integers of the form 4m + 2. Doubly even numbers are those with ν2(n) > 1, i.e., integers of the form 4m. 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 in pure math, although it is used in children's teaching materials including higher multiples such as "quadruply even." One can also extend the 2-order to the rational numbers by defining ν2(q) to be the unique integer ν where
q = 2 ν a b {\displaystyle q=2^{\nu }{\frac {a}{b}}}
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
| n | 2 = 2 − ν 2 ( n ) , {\displaystyle |n|_{2}=2^{-\nu _{2}(n)},}
one is well on the way to constructing the 2-adic numbers.
Applications
Safer outs in darts The object of the game of darts is to reach a score of 0, so the player with the smaller score is in a better position to win. At the beginning of a leg, "smaller" has the usual meaning of absolute value, and the basic strategy is to aim at high-value areas on the dartboard and score as many points as possible. At the end of a leg, since one needs to double out to win, the 2-adic absolute value becomes the relevant measure. With any odd score no matter how small in absolute value, it takes at least two darts to win. Any even score between 2 and 40 can be satisfied with a single dart, and 40 is a much more desirable score than 2, due to the effects of missing. A common miss when aiming at the double ring is to hit a single instead and accidentally halve one's score. Given a score of 22 — a singly even number — one has a game shot for double 11. If one hits single 11, the new score is 11, which is odd, and it will take at least two further darts to recover. By contrast, when shooting for double 12, one may make the same mistake but still have 3 game shots in a row: D12, D6, and D3. Generally, with a score of n < 42, one has ν2(n) such game shots. This is why 32 = 25 is such a desirable score: it splits 5 times.
Irrationality of the square root of 2 The classic proof that the square root of 2 is irrational operates by infinite descent. Usually, the descent part of the proof is abstracted away by assuming (or proving) the existence of irreducible representations of rational numbers. An alternate approach is to exploit the existence of the ν2 operator. Assume by contradiction that
2 = a b , {\displaystyle {\sqrt {2}}={\frac {a}{b}},}
where a and b are non-zero natural numbers. Square both sides of the equality and apply the 2-order valuation operator ν2 to 2b2 = a2:
ν 2 ( 2 b 2 ) = ν 2 ( a 2 ) {\displaystyle \nu _{2}\left(2b^{2}\right)=\nu _{2}\left(a^{2}\right)}
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