# 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 35

- 35 = 2
^{3} + 3^{3}, i.e., the sum of the cubes of the first two primes
- There are 35 free (up to reflections and rotations) hexominoes, the polyominoes made from 6 squares
- 35 is the number of intersections of diagonals inside a regular heptagon
- The longest uncrossed knight's tour on a chessboard has 35 moves

## Rare Properties of 35

A *tetrahedral* number is the number of balls you can put in a triangular pyramid.

This is the space generalization of triangular and square numbers.

## Common Properties of 35

A positive integer greater than 1 that is not prime is called *composite*.

Composite numbers are opposite to prime numbers.

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 *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 number is said to be *square-free* if its prime decomposition contains no repeated factors.