# 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 64

- 64 is the number of squares on a chessboard
- 64 is the smallest power of 2 with no prime neighbor
- 64 is the smallest non-trivial square and cube at the same time
- 64 is alphanumeric value of ZERO
- 64 is the difference of the first pair of amicable numbers: 284 − 220

## Rare Properties of 64

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

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 64

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 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.