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{{Infobox number
| number = 126
| divisor = 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
}}
'''126''' ('''one hundred [and] twenty-six''') is the [[natural number]] following [[125 (number)|125]] and preceding [[127 (number)|127]].

==In mathematics==
As the [[binomial coefficient]] <math>\tbinom{9}{4}</math>, 126 is a [[central binomial coefficient]]<ref>{{SloanesRef|A001405|name=Central binomial coefficients}} See also [[OEIS:A001700]] for the odd central binomial coefficients.</ref> and a [[pentatope number]].<ref>{{citation|title=Figurate Numbers|first1=Elena|last1=Deza|first2=M.|last2=Deza|author2-link=Michel Deza|publisher=World Scientific|year=2012|isbn=9789814355483|page=162|contribution=3.1 Pentatope numbers and their multidimensional analogues}}; {{SloanesRef|A000332|name=Binomial coefficients binomial(n,4)}}</ref> It is also a [[decagonal number]]<ref>{{harvtxt|Deza|Deza|2012}}, pp. 2–3 and 6; {{SloanesRef|A001107|name=10-gonal (or decagonal) numbers}}</ref> and a [[pentagonal pyramidal number]].<ref>{{harvtxt|Deza|Deza|2012}}, pp. 93, 211. {{SloanesRef|A002411|name=Pentagonal pyramidal numbers}}</ref> As 125&nbsp;+&nbsp;1 it is &sigma;<sub>3</sub>(5), the fifth value of the [[Divisor function|sum of cubed divisors function]],<ref>{{SloanesRef|A001158|name=sigma_3(n): sum of cubes of divisors of n}}</ref> and is a sum of two cubes.<ref>{{SloanesRef|A003325|name=Numbers that are the sum of 2 positive cubes}}</ref>

There are exactly 126 crossing points among the diagonals of a regular [[nonagon]],<ref>{{SloanesRef|A006561|name=Number of intersections of diagonals in the interior of regular n-gon}}</ref> 126 [[binary string]]s of length seven that are not repetitions of a shorter string,<ref>{{SloanesRef|A027375|name=Number of aperiodic binary strings of length n; also number of binary sequences with primitive period n}}</ref> 126 different [[semigroup]]s on four elements (up to isomorphism and reversal),<ref>{{SloanesRef|A001423|name=Number of semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator)}}</ref> and 126 different ways to partition a [[decagon]] into even polygons by diagonals.<ref>{{SloanesRef|A003168|name=Number of blobs with 2n+1 edges}}</ref> There are exactly 126 positive integers that are not solutions of the equation
:<math>x=abc+abd+acd+bcd,</math>
where ''a'', ''b'', ''c'', and ''d'' must themselves all be positive integers.<ref>{{SloanesRef|A027566|name=Number of numbers not of form k_1 k_2 .. k_n (1/k_1 + .. + 1/k_n), k_i >= 1}}. See [[OEIS:A027563]] for the list of these 126 numbers.</ref>

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.<ref>J. D. Koninck, Those Fascinating Numbers, transl. author. American Mathematical Society (2008) p. 40.</ref>

==In physics==
126 is is the seventh [[Magic number (physics)|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.<ref>{{citation|title=Nature's Building Blocks: An A-Z Guide to the Elements|first=John|last=Emsley|publisher=Oxford University Press|year=2011|isbn=9780199605637|page=592|url=http://books.google.com/books?id=4BAg769RfKoC&pg=PA592}}</ref>

== See also ==
* [[126th (disambiguation)]]
* The years [[126 AD]] and [[126 BC]]
* [[126 Velleda]], a [[Main belt]] [[asteroid]]
* [[List of highways numbered 126]]
* [[126 film]] and [[126 film (roll format)]] photographic film formats

==References==
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{{Integers|1}}

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[[Category:Integers]]