# 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 4523

- 4523 is the earliest 4-digit prime to appear in the decimal expansion of e

## Rare Properties of 4523

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.

- 2,
- 71,
- 271,
**4523**, - 74713,
- 904523,
- 2718281,
- ...

## Common Properties of 4523

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

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 *odd* if it is not divisible by 2.

Numbers that are not odd are even. 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.

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.