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 K_{5}, 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 10^{10000000000}.

• Giovanni Resta, 2013 • e-mail: info -at- numbersaplenty.com • Privacy notice • engine limits •