# 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 91

- 91 is the smallest positive integer expressible as a sum of two cubes in two different ways if negative cubes are allowed: 91 = 6
^{3} + (-5)^^{3} = 4^{3} + 3^{3}
- 91 is the first non-trivial composite: every smaller composite is either even, a familiar square, ends in 5, has a digit sum that is a multiple of 3, or is obviously divisible by 11
- In cents of a U.S. dollar, 91 is the amount of money one has if one has one each of the coins of denominations less than a dollar (penny, nickel, dime, quarter and half dollar)
- 91 is the smallest pseudoprime to base 3
- 91 is the smallest Cuban composite, that is the smallest composite number equal to the difference of two consecutive cubes
- 91 is the smallest number which is both the sum and the difference of two consecutive positive cubes (submitted by Anthony Kozar)

## Common Properties of 91

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.

One can take the sum of the squares of the digits of a number. Those numbers are *happy* for which iterating this operation eventually leads to 1.

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.

If you start with n points on a line, then draw n-1 points above and between, then n-2 above and between them, and so on, you will get a triangle of points. The number of points in this triangle is a *triangle* number.

Compare to square, pentagonal and tetrahedral numbers.