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 512
- 512 is the smallest power of 2 of the form 2n such that both 2n - n2 and 2n + n2 are prime
- 512 is the smallest integer (except 1) which is the cube of the sum of its digits
- 512 is the smallest cube that is a sum of a positive cube and a positive square (512 = 169 + 343)
- 512 is the only 3-digit number that equals the sum of its digits to the power of the number of its digits
Rare Properties of 512
The number n is a cube if it is the cube of an integer.
A number is a power of 2 if it is 2 to some power.
Common Properties of 512
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 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.
An integer n is powerful if for every prime p dividing n, p2 also divides n.
How much power? They all can be written as a2 b3.
The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.