# 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 131

- The 131st Fibonacci number (1066340417491710595814572169) is the smallest Fibonacci prime which contains all the digits from 0 to 9
- 131 is the smallest number n such that n + 1 and n − 1 each has exactly 3 distinct prime factors
- 131 is the smallest Honaker prime: the sum of its digits equals the sum of the digits of its index (32) in the prime sequence
- 131 is the smallest prime that is a concatenation of a prime and its first digit
- 131 is the smallest palindromic prime made of only two digits such that swapping those digits creates another prime
- 131 is the smallest prime which stays prime when the end digits (on both sides) are repeated once (11311 is also prime)
- 131 is the smallest mountain prime
- 131 is the smallest palindromic prime with three primes embedded in it (13, 3 and 31)

## Common Properties of 131

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.

The next *Ulam* number is uniquely the sum of two earlier distinct Ulam numbers.

*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.