Even and odd numbers in the context of "Sequence (mathematics)"

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an arbitrary index set.

For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...).

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.

The conjecture has been shown to hold for all natural numbers less than 4×10, but remains unproven despite considerable effort.

In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:

with x and y integers, if and only if

10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.

The number "ten" originates from the Proto-Germanic root "*tehun", which in turn comes from the Proto-Indo-European root "*dekm-", meaning "ten". This root is the source of similar words for "ten" in many other Germanic languages, like Dutch, German, and Swedish. The use of "ten" in the decimal system is likely because humans have ten fingers and ten toes, which people may have used to count by.

In nuclear physics, properties of a nucleus depend on evenness or oddness of its atomic number (proton number) Z, neutron number N and, consequently, of their sum, the mass number A. Most importantly, oddness of both Z and N tends to lower the nuclear binding energy, making odd nuclei generally less stable. This effect is not only experimentally observed, but is included in the semi-empirical mass formula and explained by some other nuclear models, such as the nuclear shell model. This difference of nuclear binding energy between neighbouring nuclei, especially of odd-A isobars, has important consequences for beta decay.

The nuclear spin is zero for even-Z, even-N nuclei, integer for all even-A nuclei, and odd half-integer for all odd-A nuclei.