# 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 11

- 11 is the smallest prime such that 2
^{p}-1 is not prime
- 11 is the largest number which is not expressible as the sum of two composite numbers
- 11 is the smallest prime for which the sum of digits equals the number of digits
- 11 is the only prime comprising an even number of identical digits
- 11 is the smallest strobogrammatic prime
- 11 is the only palindromic prime with an even number of digits
- 11 is the only prime whose period length is two
- 11 divides all palindromes with an even number of digits
- The diagonals of a regular pentagon divide it into 11 regions
- 11 is the number of 2 by 2 binary matrices M such that M
^{2} is also a binary matrix

## Rare Properties of 11

A *repunit* is an integer in which every digit is one.

The term repunit comes from combining "repeated" and "unit".

## Common Properties of 11

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 n-th *lazy caterer* number is the maximum number of pieces a (circular) pizza can be cut into with n (straight-line) cuts.

Unlike the situation with cake, everybody gets the toppings.

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.

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