# 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
- If the Goldbach's conjecture is true, then 5 is the only odd untouchable number
- 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 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) n^{2} 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 -- 10^{k} + 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.