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{{Short description|How many times a number is divisible by 2}}

In [[mathematics]] an [[even integer]], that is, a number that is [[divisibility|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 value ''k'' such that ''n''/2<sup>''k''</sup> is an integer. This is equivalent to the [[multiplicity (mathematics)|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 code]]s), among others.

==Definitions==
The ancient Greek terms "even-times-even" ({{langx|grc|ἀρτιάκις ἄρτιος}}) and "even-times-odd" ({{langx|grc|ἀρτιάκις περισσός}} or {{lang|grc|ἀρτιοπέριττος}}) were given various inequivalent definitions by [[Euclid]] and later writers such as [[Nicomachus]].<ref>{{cite book |title=The Thirteen Books of Euclid's Elements |author=Euclid; Johan Ludvig Heiberg |year=1908 |publisher=The University Press |pages=[https://archive.org/details/bub_gb_lxkPAAAAIAAJ/page/n285 281]–284 |url=https://archive.org/details/bub_gb_lxkPAAAAIAAJ}}</ref> 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|''p''-adic order]] at a general [[prime number]] ''p''; see [[p-adic number|''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 2<sup>ν</sup> [[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]]).<ref>{{cite journal |first=Tamas |last=Lengyel |title=Characterizing the 2-adic order of the logarithm |journal=The Fibonacci Quarterly |volume=32 |year=1994 |issue=5 |pages=397–401 |doi=10.1080/00150517.1994.12429184 |url=http://employees.oxy.edu/lengyel/papers/FQ/pdf/log2.pdf}}</ref> The 2-order of ''n'' is written ν<sub>2</sub>(''n'') or ord<sub>2</sub>(''n''). It is not to be confused with the multiplicative [[order (group theory)|order]] [[multiplicative group of integers modulo n|modulo 2]].

The 2-order provides a unified description of various classes of integers defined by evenness:
*Odd numbers are those with ν<sub>2</sub>(''n'') = 0, i.e., integers of the form {{nowrap|2''m'' + 1}}.
*Even numbers are those with ν<sub>2</sub>(''n'') > 0, i.e., integers of the form {{nowrap|2''m''}}. In particular:
**Singly even numbers are those with ν<sub>2</sub>(''n'') = 1, i.e., integers of the form {{nowrap|4''m'' + 2}}.
**Doubly even numbers are those with ν<sub>2</sub>(''n'') > 1, i.e., integers of the form {{nowrap|4''m''}}.
***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."<ref>url=https://www.parleybot.com/p/double-triple-quadruple-even-number.html | Multiple-evens Online Calculator</ref>

One can also extend the 2-order to the [[rational numbers]] by defining ν<sub>2</sub>(''q'') to be the unique integer ν where
:<math>q = 2^\nu\frac{a}{b}</math>
and ''a'' and ''b'' are both odd. For example, [[half-integer]]s have a negative 2-order, namely &minus;1. Finally, by defining the 2-adic absolute value
:<math>|n|_2 = 2^{-\nu_2(n)},</math>
one is well on the way to constructing the [[p-adic number|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 {{nowrap|''n'' < 42}}, one has {{nowrap|ν<sub>2</sub>(''n'')}} such game shots. This is why {{nowrap|1=32 = 2<sup>5</sup>}} is such a desirable score: it splits 5 times.<ref>{{cite book |title=Children Doing Mathematics |url=https://archive.org/details/childrendoingmat00nune |url-access=limited |author=[[Terezinha Nunes|Nunes, Terezinha]] and Peter Bryant |year=1996 |publisher=Blackwell |isbn=0-631-18472-4 |pages=[https://archive.org/details/childrendoingmat00nune/page/n115 98]–99}}</ref><ref>{{cite book |title=A Bar Player's Guide to Winning Darts |first=Fred |last=Everson |year=2006 |publisher=Trafford |isbn=1-55369-321-3 |page=39}}</ref>

===Irrationality of the square root of 2===
The classic proof that the [[square root of 2]] is [[irrational number|irrational]] operates by [[infinite descent]]. Usually, the descent part of the proof is abstracted away by assuming (or proving) the existence of [[irreducible fraction|irreducible]] representations of [[rational number]]s. An alternate approach is to exploit the existence of the ν<sub>2</sub> operator.

[[Assume by contradiction]] that

:<math>\sqrt 2 = \frac a b,</math>

where ''a'' and ''b'' are non-zero natural numbers. Square both sides of the equality and apply
the 2-order valuation operator ν<sub>2</sub> to {{nowrap|1=2''b''<sup>2</sup> = ''a''<sup>2</sup>}}:

:<math>\nu_2\left(2b^2\right) = \nu_2\left(a^2\right)</math>
:<math>\nu_2\left(b^2\right) + 1 = \nu_2\left(a^2\right)</math>
:<math>2\nu_2(b) + 1 = 2\nu_2(a)</math>
:<math>\nu_2(a) - \nu_2(b) = \frac 1 2 </math>

Since 2-order valuations are integers, the difference cannot be equal to the rational <math display="inline">\frac 1 2</math>. By contradiction, therefore, {{radic|2}} is not a rational. 

More concretely, since the valuation of 2''b''<sup>2</sup> is odd, while valuation of ''a''<sup>2</sup> is even, they must be distinct integers, so that <math>\left|2 b^2 - a^2\right| \geq 1</math>.  An easy calculation then yields a lower bound of <math display="inline">\frac{1}{3b^2}</math> for the difference <math>\left|\sqrt 2 - a / b \right|</math>, yielding a direct proof of irrationality not relying on the [[law of excluded middle]].<ref>{{cite book |title=The Moment of Proof: Mathematical Epiphanies |first=Donald C. |last=Benson |year=2000 |publisher=Oxford UP |isbn=0-19-513919-4 |pages=46–47}}</ref>

===Geometric topology===
In [[geometric topology]], many properties of manifolds depend only on their dimension mod 4 or mod 8; thus one often studies manifolds of singly even and doubly even dimension (4''k''+2 and 4''k'') as classes. For example, doubly even-dimensional manifolds have a ''symmetric'' [[nondegenerate bilinear form]] on their middle-dimension [[cohomology group]], which thus has an integer-valued [[signature (topology)|signature]]. Conversely, singly even-dimensional manifolds have a [[Skew-symmetric graph|''skew''-symmetric]] nondegenerate bilinear form on their middle dimension; if one defines a [[quadratic refinement]] of this to a [[quadratic form]] (as on a [[framed manifold]]), one obtains the [[Arf invariant]] as a mod 2 invariant. Odd-dimensional manifolds, by contrast, do not have these invariants, though in [[algebraic surgery theory]] one may define more complicated invariants. This 4-fold and 8-fold periodicity in the structure of manifolds is related to the 4-fold periodicity of [[L-theory]] and the 8-fold periodicity of real [[topological K-theory]], which is known as [[Bott periodicity]].

If a [[compact space|compact]] [[oriented manifold|oriented]] [[smooth manifold|smooth]] [[spin manifold]] has dimension {{nowrap|''n'' ≡ 4 mod 8}}, or {{nowrap|1=ν<sub>2</sub>(''n'') = 2}} exactly, then its [[signature (topology)|signature]] is an integer multiple of 16.<ref>Ochanine, Serge, "Signature modulo 16, invariants de Kervaire généralisés et nombres caractéristiques dans la K-théorie réelle", Mém. Soc. Math. France 1980/81, no. 5, 142 pp.  {{MathSciNet|id=1809832}}</ref>

===Other appearances===

A singly even number cannot be a [[powerful number]]. It cannot be represented as a [[difference of two squares]]. However, a singly even number can be represented as the difference of two [[pronic number]]s or of two powerful numbers.<ref group=>* {{cite journal  | author = McDaniel, Wayne L.  | title = Representations of every integer as the difference of powerful numbers  | journal = [[Fibonacci Quarterly]]  | volume = 20  | year = 1982  | pages = 85–87| doi = 10.1080/00150517.1982.12430037 }}</ref>

In [[group theory]], it is relatively simple<ref>See, for example: {{cite book |title=Elements of mathematics: Algebra I: Chapters 1-3 |author=Bourbaki |publisher=Springer |year=1989 |edition=Softcover reprint of 1974 English translation |isbn=3-540-64243-9 |pages=154–155}}</ref> to show that the order of a [[nonabelian group|nonabelian]] [[finite simple group]] cannot be a singly even number. In fact, by the [[Feit–Thompson theorem]], it cannot be odd either, so every such group has doubly even order.

[[Lambert's continued fraction]] for the [[tangent function]] gives the following [[simple continued fraction]] involving the positive singly even numbers:<ref>{{cite book |author=Hairer, Ernst and Gerhard Wanner |title=Analysis by Its History |url=https://archive.org/details/analysisbyitshis0000hair |url-access=registration |year=1996 |publisher=Springer |isbn=0-387-94551-2 |pages=[https://archive.org/details/analysisbyitshis0000hair/page/69 69–78]}}</ref>

:<math>\tanh \frac{1}{2} = \frac{e - 1}{e + 1} = 0 + \cfrac{1}{2 + \cfrac{1}{6 + \cfrac{1}{10 + \cfrac{1}{14 + \cfrac{1}{\ddots}}}}}</math>

This expression leads to similar [[List of representations of e|representations of {{math|''e''}}]].<ref>{{cite book |title=Introduction to Diophantine Approximations |first=Serge |last=Lang |year=1995 |publisher=Springer |isbn=0-387-94456-7 |pages=69–73}}</ref>

In [[organic chemistry]], [[Hückel's rule]], also known as the 4n + 2 rule, predicts that a [[cyclic compound|cyclic]] [[pi bond|π-bond]] system containing a singly even number of [[electron configuration|p electron]]s will be [[aromatic]].<ref>{{cite book |title=Organic Chemistry |author=Ouellette, Robert J. and J. David Rawn |publisher=Prentice Hall |isbn=0-02-390171-3 |page=473 |year=1996}}</ref>

==Related classifications==
Although the 2-order can detect when an integer is congruent to 0 (mod 4) or 2 (mod 4), it cannot tell the difference between 1 (mod 4) or 3 (mod 4). This distinction has some interesting consequences, such as [[Fermat's theorem on sums of two squares]].

==See also==

* [[p-adic order]]

==References==
{{reflist}}

==External links==
*[http://planetmath.org/encyclopedia/SinglyEvenNumber.html singly even number] at [[PlanetMath]]
*{{OEIS el|sequencenumber=A016825|name=Numbers congruent to 2 mod 4|formalname=Positive integers congruent to 2 mod 4: a(n) = 4n+2, for n >= 0}}
*{{OEIS el|sequencenumber=A008586|name=Multiples of 4}}

[[Category:Integer sequences]]
[[Category:Parity (mathematics)]]
[[Category:Elementary number theory]]