# 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 16

- 16 is the number of vertices of a tesseract
- 16 is the only number of the form x
^{y}=y^{x} with different x and y
- 16 is the smallest prime power of a prime power of a prime
- 16 is the base of the hexadecimal number system, which is used extensively in computer science
- 16 is the smallest short leg in a primitive Pythagorean triangle whose three side lengths are composite integers
- 16 is the smallest number which is the sum of two distinct odd primes in two ways: 16 = 3 + 13 = 5 + 11
- 16 is the only 2-digit number n such that n
^{n} ends with two copies of n: 16^{16} = 18446744073709551616
- 16 is the largest number n such that every set of n consecutive integers contain a number which is relatively prime to all the others

## Rare Properties of 16

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

The number n is a *square* if it is the square of an integer.

## Common Properties of 16

A positive integer greater than 1 that is not prime is called *composite*.

Composite numbers are opposite to prime numbers.

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.

An integer n is *powerful* if for every prime p dividing n, p^{2} also divides n.

How much power? They all can be written as a^{2} b^{3}.

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

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