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 2p-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 M2 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.