126 (one hundred ndtwenty-six) is the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

following 125 and preceding 127.

In mathematics

As the

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

\tbinom

, 126 is a

central binomial coefficient In mathematics the ''n''th central binomial coefficient is the particular binomial coefficient : = \frac = \prod\limits_^\frac \textn \geq 0. They are called central since they show up exactly in the middle of the even-numbered rows in Pascal ...

and a pentatope number. It is also a

decagonal number A decagonal number is a figurate number that extends the concept of triangular and square numbers to the decagon (a ten-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of decagonal number ...

, a Harshad number and a pentagonal pyramidal number. As 125 + 1 it is σ₃(5), the fifth value of the sum of cubed divisors function, and is a sum of two cubes. There are exactly 126 crossing points among the diagonals of a regular nonagon, 126 binary strings of length seven that are not repetitions of a shorter string, 126 different

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

s on four elements (up to isomorphism and reversal), and 126 different ways to partition a

decagon In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. A self-intersecting ''regular decagon'' i ...

into even polygons by diagonals. There are exactly 126 positive integers that are not solutions of the equation :

x=abc+abd+acd+bcd,

where ''a'', ''b'', ''c'', and ''d'' must themselves all be positive integers. It is the fifth

Granville number In mathematics, specifically number theory, Granville numbers, also known as \mathcal-perfect numbers, are an extension of the perfect numbers. The Granville set In 1996, Andrew Granville proposed the following construction of a set Set, The Se ...

, and the third such not to be a perfect number. Also, it is known to be the smallest Granville number with three distinct prime factors, and perhaps the only such Granville number.

In physics

126 is the seventh magic number in

nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies t ...

. For each of these numbers, 2, 8, 20, 28, 50, 82, and 126, an atomic nucleus with this many protons is or is predicted to be more stable than for other numbers. Thus, although there has been no experimental discovery of element 126, tentatively called unbihexium, it is predicted to belong to an island of stability that might allow it to exist with a long enough

half life Half-life (symbol ) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable ...

that its existence could be detected.

References

{{DEFAULTSORT:126 (Number) Integers

In mathematics

In physics

See also

References