# 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 181

- 181 is the smallest multi-digit palindromic prime with an odd number of digits, such that increasing the middle digit by 1 produces another prime
- 181 is the start of the first occurrence of two successive primes (181, 191) using no prime digits
- 181 starts the first occurrence of two consecutive primes which are both multidigit palindromes: 181, 191
- 181 is the smallest palindromic prime with digit sum 10
- 181 is the smallest number n such that the binary complement of n is greater than the binary complement of n
^{2}
- 181 is the smallest number that have more zeros in its binary representation compared to its square

## Common Properties of 181

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 *palindrome* is a number that reads the same forward or backward.

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.

A prime number is called a *twin* prime if there exists another prime number differing from it by 2.

*Undulating* numbers are numbers of the form abababab... in base 10.

This property is significant starting from 3-digit numbers, so we will not consider numbers below 100.