# 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 2

- 2 is the smallest prime number
- 2 is the only even prime number
- 2 is the only number whose factorial is prime
- For any polyhedron, 2 is the number of vertices plus the number of faces minus the number of edges
- The smallest field has 2 elements
- 2 is the only number that it is its own primorial, as well as its own factorial
- 2 is the only number that isn't n-polygonal with any n (submitted by Alexey Radul)
- 2 is the largest deficient factorial (submitted by Alexey Radul)

## Rare Properties of 2

The n-th *cake* number is the maximum number of pieces a (cylindrical) cake can be cut into with n (straight-plane) cuts.

Unfortunately, not everybody gets the frosting. If you cut pizza rather than cake, you get lazy caterer's numbers.

The n-th *Catalan* number is equal to (2n choose n)/(n+1) = (2n)!/(n!(n+1)!).

There are many ways Catalan numbers can be interpreted; there are some cool pictures here and the Wikipedia article is very good.

The n-th *factorial* is the product of the first n natural numbers.

The factorial deserved an exclamation mark for its notation: k! = 1*2*3*...*k.

*Fibonacci* numbers are numbers that form the Fibonacci sequence. The Fibonacci sequence is defined as starting with 1, 1 and then each next term is the sum of the two preceding ones.

Fibonacci numbers are very common in nature. For example, a pineapple has 8 spirals if you count one way, and 13 if you count the other way.

The n-th *Google* number is the first n-digit prime found in the decimal expansion of e.

They are named *Google* numbers because of the unusual hiring ad that *Google* put up.

A k-digit number n is called *narcissistic* if it is equal to the sum of k-th powers of its digits. They are also called *Plus Perfect* numbers.

A number is a *power of 2* if it is 2 to some power.

The p-*primorial* is the product of all primes less than or equal to p. It is sometimes denoted by p#.

Compare to compositorials and factorials.

The number is called *pronic* if it is the product of two consecutive numbers.

They are twice triangular numbers.

## Common Properties of 2

The number n is *deficient* if the sum of all its positive divisors except itself is less than n.

Compare with perfect and abundant numbers.

A number is *even* if it is divisible by 2.

Numbers that are not even are odd. Compare with another pair -- evil and odious numbers.

The n-th *lazy caterer* number is the maximum number of pieces a (circular) pizza can be cut into with n (straight-line) cuts.

Unlike the situation with cake, everybody gets the toppings.

The number n is *odious* if it has an odd number of 1's in its binary expansion.

Guess what evil numbers are.

A *palindrome* is a number that reads the same forward or backward.

The number n is *practical* if all numbers strictly less than n are sums of distinct divisors of n.

A *prime* is a positive integer greater than 1 that is divisible by no positive integers other than 1 and itself.

Prime numbers are opposite to composite numbers.

A number is said to be *square-free* if its prime decomposition contains no repeated factors.

The next *Ulam* number is uniquely the sum of two earlier distinct Ulam numbers.

The *untouchable* numbers are those that are not the sum of the proper divisors of any number.