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 2
- 2 is the smallest prime number
- 2 is the only even prime number
- 2 is the only number whose factorial is prime
- For any polyhedron, 2 is the number of vertices plus the number of faces minus the number of edges
- The smallest field has 2 elements
- 2 is the only number that it is its own primorial, as well as its own factorial
- 2 is the only number that isn't n-polygonal with any n (submitted by Alexey Radul)
- 2 is the largest deficient factorial (submitted by Alexey Radul)
- 2 is the only prime which does not contain the letter E once spelled out (submitted by Aaron Diveley)
- 2 is the only difference between two consecutive primes which is prime
- 2 is the only number such that if you add it to itself or multiply by itself you would get the same number
- 2 is the only prime such that the sum of its divisors is another prime
- 2 is the only prime that lack letter e in its English name
Rare Properties of 2
The n-th cake number is the maximum number of pieces a (cylindrical) cake can be cut into with n (straight-plane) cuts.
Unfortunately, not everybody gets the frosting. If you cut pizza rather than cake, you get lazy caterer's numbers.
The n-th Catalan number is equal to (2n choose n)/(n+1) = (2n)!/(n!(n+1)!).
There are many ways Catalan numbers can be interpreted; there are some cool pictures here and the Wikipedia article is very good.
The n-th factorial is the product of the first n natural numbers.
The factorial deserved an exclamation mark for its notation: k! = 1*2*3*...*k.
Fibonacci numbers are numbers that form the Fibonacci sequence. The Fibonacci sequence is defined as starting with 1, 1 and then each next term is the sum of the two preceding ones.
Fibonacci numbers are very common in nature. For example, a pineapple has 8 spirals if you count one way, and 13 if you count the other way.
The n-th Google number is the first n-digit prime found in the decimal expansion of e.
They are named Google numbers because of the unusual hiring ad that Google put up.
A k-digit number n is called narcissistic if it is equal to the sum of k-th powers of its digits. They are also called Plus Perfect numbers.
A number is a power of 2 if it is 2 to some power.
The p-primorial is the product of all primes less than or equal to p. It is sometimes denoted by p#.
Compare to compositorials and factorials.
The number is called pronic if it is the product of two consecutive numbers.
They are twice triangular numbers.
Common Properties of 2
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 even if it is divisible by 2.
Numbers that are not even are odd. Compare with another pair -- evil and odious 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.
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.
The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.
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.
The untouchable numbers are those that are not the sum of the proper divisors of any number.