# Number Gossip

(Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.)

## Unique Properties of 1

- 1 is the multiplicative identity
- 1 is the only positive integer that is neither prime nor composite
- 1 is its own factorial, and its own square and cube (and so on, as 1 * 1 * ... * 1 = 1)
- 1 is the only number whose concatenation with itself can yield primes in many cases
- 1 is the only number with exactly one positive divisor
- 1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number to name just a few
- 1 is the only square-free square (submitted by Alexey Radul)
- 1 is the only triangular cube (proved by Max Alekseyev and Jaap Spies)