# 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 1423

- 1423 is the smallest prime number that can be formed by concatenating the first several positive integers
- 1423 is the smallest prime containing 2n digits in which the sum of the first n digits equals that of the rest
- 1423 is the smallest prime with nonzero digits such that sum of the cubes of its digits equals the square of the sum of its digits (1
^{3}+4^{3}+2^{3}+3^{3} = (1+4+2+3)^{2})

## Common Properties of 1423

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.