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)
  • In English, one is the only number whose constituent letters appear in reverse alphabetical order
  • 1 is the only number that divides every other number
  • 1 is the only number that no other number divides

Rare Properties of 1

Automorphic (Curious)
Catalan
Cube
Factorial
Fibonacci
Narcissistic
Pentagonal
Power of 2
Repunit
Square
Tetrahedral (Pyramidal)

Common Properties of 1

Deficient
Happy
Lucky
Odd
Odious
Palindrome
Powerful
Practical
Square-free
Triangular
Ulam