# 126 (number)

Short description: Natural number
 ← 125 126 127 →
Cardinalone hundred twenty-six
Ordinal126th
(one hundred twenty-sixth)
Factorization2 × 32 × 7
Divisors1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
Greek numeralΡΚϚ´
Roman numeralCXXVI
Binary11111102
Ternary112003
Quaternary13324
Quinary10015
Senary3306
Octal1768
DuodecimalA612
Vigesimal6620
Base 363I36

126 (one hundred [and] twenty-six) is the natural number following 125 and preceding 127.

## In mathematics

As the binomial coefficient $\displaystyle{ \tbinom{9}{4} }$, 126 is a central binomial coefficient[1] and a pentatope number.[2] It is also a decagonal number[3] and a pentagonal pyramidal number.[4] As 125 + 1 it is σ3(5), the fifth value of the sum of cubed divisors function,[5] and is a sum of two cubes.[6]

There are exactly 126 crossing points among the diagonals of a regular nonagon,[7] 126 binary strings of length seven that are not repetitions of a shorter string,[8] 126 different semigroups on four elements (up to isomorphism and reversal),[9] and 126 different ways to partition a decagon into even polygons by diagonals.[10] There are exactly 126 positive integers that are not solutions of the equation

$\displaystyle{ x=abc+abd+acd+bcd, }$

where a, b, c, and d must themselves all be positive integers.[11]

It is the fifth Granville number, 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.[12]

## In physics

126 is the seventh magic number in nuclear physics. 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 that its existence could be detected.[13]