# 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 8911

- 8911 is the largest 4-digit pseudoprime to base 2
- 8911 is the smallest Carmichael number that can't be represented as a sum of three squares
- 8911 is the smallest Carmichael number all of whose prime factors are congruent to 3 modulo 4
- 8911 is the smallest odd quarefree composite numbers n such that p-1 divides n-1 and p-1 does not divide (n-1)/2 for every prime factor of n

## Rare Properties of 8911

The composite integer n is a *Carmichael* number if b^{n-1} = 1 (mod n) for every integer b
which is relatively prime with n.

Carmichael numbers behave like prime numbers with respect to the most useful primality test, that is they pretend to be prime.

## Common Properties of 8911

The number n is called an *apocalyptic power* if 2^{n} contains the consecutive digits 666 (in decimal).

A positive integer greater than 1 that is not prime is called *composite*.

Composite numbers are opposite to prime numbers.

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 number is said to be *square-free* if its prime decomposition contains no repeated factors.

If you start with n points on a line, then draw n-1 points above and between, then n-2 above and between them, and so on, you will get a triangle of points. The number of points in this triangle is a *triangle* number.

Compare to square, pentagonal and tetrahedral numbers.