# 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 101

- 101 is the smallest naught-y prime, that is a prime with noughts (or zeros)
- 101 is the first odd prime for which the Mertens function is 0
- 101 is the sum of all products p*q involving the four first primes: 101: 2*3 + 2*5 + 2*7 + 3*5 + 3*7 + 5*7
- 101 is the smallest odd prime which can be represented as the sum of a square and its reverse, i.e., 100 + 001 = 101
- 101 is the smallest prime whose sum of the digits is less than the number of digits
- 101 is the largest prime factor of the smallest palindromic Carmichael number (101101)
- 101 is the only prime of the form 1010...101
- 101 is the smallest three-digit prime whose digits are not prime
- 101 is the smallest 3-digit palindromic prime
- 101 is the smallest undulating prime

## Common Properties of 101

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 *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.