Singly and doubly even

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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.

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[edit] 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.[1] 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 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 evenness of zero).[2] 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.

One can also extend the 2-order to the rational numbers by defining ν2(q) to be the unique integer ν where

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 norm,

|n|_2 = 2^{-\nu_2(n)},

one is well on the way to constructing the 2-adic numbers.

[edit] Applications

[edit] 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 norm 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.[3][4]

[edit] Irrationality 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

\sqrt2 = a / b,

where a and b do not have to be in lowest terms. Then applying ν2 to the equation 2b2 = a2 yields[5]

\frac12 = \nu_2(a) - \nu_2(b),

which is absurd. Therefore √2 is irrational.

[edit] 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 numbers or of two powerful numbers.

In group theory, it is relatively simple[6] to show that the order of a 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 continued fraction involving the positive singly even numbers:[7]

\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}}}}}

This expression leads to similar representations of e.[8]

If a compact oriented smooth spin manifold has dimension n ≡ 4 mod 8, or ν2(n) = 2 exactly, then its signature is an integer multiple of 16.[9]

In organic chemistry, Hückel's rule, also known as the 4n + 2 rule, predicts that a cyclic π-bond system containing a simply even number of p electrons will be aromatic.[10]

[edit] 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.

[edit] References

  1. ^ Euclid; Johan Ludvig Heiberg (1908). The Thirteen Books of Euclid's Elements. The University Press, pp. 281-284. 
  2. ^ Lengyel, Tamas (1994). "Characterizing the 2-adic order of the logarithm". The Fibonacci Quarterly 32: 397–401. 
  3. ^ Nunes, Terezinha and Peter Bryant (1996). Children Doing Mathematics. Blackwell, 98-99. ISBN 0631184724. 
  4. ^ Everson, Fred (2006). A Bar Player's Guide to Winning Darts. Trafford, 39. ISBN 1553693213. 
  5. ^ Benson, Donald C. (2000). The Moment of Proof: Mathematical Epiphanies. Oxford UP, 46-47. ISBN 0195139194. 
  6. ^ See, for example: Bourbaki (1989). Elements of mathematics: Algebra I: Chapters 1-3, Softcover reprint of 1974 English translation, Springer, 154-155. ISBN 3540642439. 
  7. ^ Hairer, Ernst and Gerhard Wanner (1996). Analysis by Its History. Springer, 69-78. ISBN 0387945512. 
  8. ^ Lang, Serge (1995). Introduction to Diophantine Approximations. Springer, 69-73. ISBN 0387944567. 
  9. ^ 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. MR1809832
  10. ^ Ouellette, Robert J. and J. David Rawn. Organic Chemistry. Prentice Hall, 473. ISBN 0-02-390171-3. 

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