Singly and doubly even

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 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/2k 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, coding theory (see even codes), among others.

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