# 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 919

- 919 is the smallest multidigit palindromic number whose sum of digits shows up as a substring of the number
- 919 is the smallest number which is not the difference between two positive palindromes
- 919 is the smallest palindromic prime whose digital root equals its middle digits
- 919 is the smallest prime p such that p-1 and p+1 are both divisible by non-trivial cubes (919 - 1 = 2*3
^{3}*17 and 919 + 1 = 2^{3}*5*23)

## Common Properties of 919

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.

*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.