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 semigroups on four elements (up to isomorphism and reversal), and 126 different ways to partition a decagon into even polygons by diagonals. There are exactly 126 positive integers that are not solutions of the equation
where a, b, c, and d must themselves all be positive integers.
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.
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.