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 x3 − y3 = z3
- 728 is the smallest number n with property that n and 2n are sums of two distinct positive cubes (728 = 63 + 83 and 1456 = 53 + 113)
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 :-) ?