22 February 2013

The remarkable number 39

This charming booklet from 1983

contains 158 pages, three of which are musical partitions and one is ruled to add your own music. The title means The remarkable numbers. In it we find the number 39 with the mention

the smallest natural number for which 
we don't know any remarkable property.

The author (mathematician and co-founder of Oulipo, read also this) is, of course, much aware that this very property makes that number remarkable; if not, it would not be in his book. He continues

The fact of being the smallest will not be considered a remarkable property, 
in order to avoid a dreadful recurrence in the rest of the book.

(Here or here the original entry in French.) We can easily extract a genuine theorem, which also implies that a book called 'The remarkable numbers' is a rip-off.

Theorem. Every natural number is remarkable.

Proof. Suppose, for contradiction, that there are natural numbers that are not remarkable. By the well-ordering property of the naturals, there would be a smallest number in the set of non-remarkable natural numbers. This smallest of non-remarkable numbers would be most remarkable, contradicting the assumption.

Corollary. There is no book containing all remarkable numbers.