# 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 157

- 157 is the smallest apocalyptic power
- 157 is the smallest prime whose binary and ternary representations are also prime when read in decimal
- 157 is the smallest prime which is the sum of ten consecutive composite numbers
- 157 is the smallest emirp whose binary conversion remains an emirp when read in decimal

## Common Properties of 157

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.