1 is neither prime nor composite

The number 1 belongs to so many families of numbers that I decided against listing them, because often the inclusion is somehow trivial.

Indeed, it belongs to most of figurate numbers families, like triangular, square or pentagonal numbers.

It is also a factorial, a powerful and a Fibonacci number, and so on and so forth...

However, there are some properties which make 1 more interesting.

For example, it is equal to the crossing number of the complete graph K5, i.e., it is possible to connect 5 points one to each other in such a way there is only one cross between edges:


Charmichael has conjectured that, for every k, the number of solutions of the equation φ(x) = k is never equal to one, i.e., either there are no solutions at all (like for k = 7) or at least two solutions (like for k = 8). K.Ford has proved that if a counterexample exists, it must be greater than 1010000000000.

to be continued...