# 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 728

- Consecutive Smith numbers are called Smith brothers: 728 is the smaller brother of the smallest pair of Smith brothers
- 728 is the smallest number n where n and n+1 are both products of 5 or more primes
- 728 is the smallest Fermat-near-miss number: it differs by 1 from a cube and it is a sum of two cubes
- 728 is the smallest number that is the sum of two (possibly negative) cubes in at least 3 ways, in other words it is the third cabtaxi number
- 728 is the smallest product of three positive integers (8,7,13) giving a solution to the Diophantine equation x
^{3} − y^{3} = z^{3}
- 728 is the smallest number n with property that n and 2n are sums of two distinct positive cubes (728 = 6
^{3} + 8^{3} and 1456 = 5^{3} + 11^{3})

## Common Properties of 728

The number n is *abundant* if the sum of all its positive divisors except itself is more than n.

They are abundant above perfection, not to mention deficiency. See perfect and deficient numbers.

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

Composite numbers are opposite to prime 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 number n is *odious* if it has an odd number of 1's in its binary expansion.

Guess what evil numbers are.

The number n is *practical* if all numbers strictly less than n are sums of distinct divisors of n.

A composite number is called a *Smith* number if the sum of its digits equals the sum of all the digits appearing in its prime divisors (counting multiplicity).

In 1984, when Albert Wilansky called his brother-in-law, named Smith, he noticed that the phone number possesses the property described here. Are they called joke numbers, because they were named after an innocent unsuspecting brother-in-law :-) ?