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 5
- 5 is the smallest number of queens needed to attack every square on a standard chess board
- 5 is the only prime which is the difference of two squares of primes
- 5 is the only prime that is a member of 2 pairs of twin primes
- 5 is the only number which is equal to the sum of all primes less than itself
- 5 is the number of Platonic solids
- 5 is the smallest degree at which polynomial roots are no longer findable in closed form
- 5 is the smallest odd prime which is not a Gaussian prime (submitted by Sam Steingold)
- 5 is the smallest number of vertices needed to create a non-planar graph
- The only polygon that has the same number of sides and diagonals is a pentagon
- 5 is the only number that is a member of two different pairs of twin primes
- 5 is the smallest number that appear in two twin Pythagorean triplets (3, 4, 5) and (5, 12, 13)
- John Conway proved that in the Conway's Soldiers game, no finite sequence of moves allows a soldier to advance to the fifth row above the horizontal line.
- The volume of a unit ball reaches its maximum in dimension 5.
Rare Properties of 5
The number n is called an automorphic number if (the decimal expansion of) n2 ends with n. These numbers are also called curious.
It is curious, how for a k-digit automorphic number n there is another automorphic number -- 10k + 1 - n. For this to work with n=1, you have to treat 1 as a zero-digit number.
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.
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 k-th hungry number is the smallest number n such that 2^n contains the first k digits of the decimal expansion of pi.
They are named hungry numbers because they try to eat as much "pi" as possible.
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.
Common Properties of 5
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 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 untouchable numbers are those that are not the sum of the proper divisors of any number.