# 126 (number)

 ← 125 126 127 →
Cardinal one hundred twenty-six
Ordinal 126th
(one hundred and twenty-sixth)
Factorization 2 × 32× 7
Divisors 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
Roman numeral CXXVI
Binary 11111102
Ternary 112003
Quaternary 13324
Quinary 10015
Senary 3306
Octal 1768
Duodecimal A612
Vigesimal 6620
Base 36 3I36

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

## In mathematics

As the binomial coefficient $\tbinom{9}{4}$, 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

$x=abc+abd+acd+bcd,$

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]

## References

1. ^ "Sloane's A001405 : Central binomial coefficients", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. See also OEIS:A001700 for the odd central binomial coefficients.
2. ^ Deza, Elena; Deza, M. (2012), "3.1 Pentatope numbers and their multidimensional analogues", Figurate Numbers, World Scientific, p. 162, ISBN 9789814355483; "Sloane's A000332 : Binomial coefficients binomial(n,4)", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. ^ Deza & Deza (2012), pp. 2–3 and 6; "Sloane's A001107 : 10-gonal (or decagonal) numbers", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
4. ^ Deza & Deza (2012), pp. 93, 211. "Sloane's A002411 : Pentagonal pyramidal numbers", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. ^
6. ^
7. ^
8. ^
9. ^
10. ^ "Sloane's A003168 : Number of blobs with 2n+1 edges", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
11. ^ "Sloane's A027566 : Number of numbers not of form k_1 k_2 .. k_n (1/k_1 + .. + 1/k_n), k_i >= 1", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.. See OEIS:A027563 for the list of these 126 numbers.
12. ^ J. D. Koninck, Those Fascinating Numbers, transl. author. American Mathematical Society (2008) p. 40.
13. ^ Emsley, John (2011), Nature's Building Blocks: An A-Z Guide to the Elements, Oxford University Press, p. 592, ISBN 9780199605637