# 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 243

- 243 is the first number of the form p
^{q} where p and q are a twin prime pair
- 243 is the smallest higher power that is a sum of two distinct higher powers (243 = 3
^{5} = 3^{3} + 6^{3})
- 243 is the smallest number n such that n and prime(n) end with the same two digits
- 243 is the only 3-digit number that is a fifth power

## Common Properties of 243

The number n is called an *apocalyptic power* if 2^{n} contains the consecutive digits 666 (in decimal).

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.

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

Guess what odious numbers are.

A number is *odd* if it is not divisible by 2.

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

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