# 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 89

- 89 is the smallest prime that is a concatenation of p
^{q} and q^{p} where p and q are prime
- 89 is the smallest prime whose digits are composites
- 89 is the only two-digit number that is the sum of its first digit and its second digit squared

## Rare Properties of 89

*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.

## Common Properties of 89

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.

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.