# 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 38

- 38 is the magic constant in the only possible magic hexagon (which utilizes all the natural integers up to and including 19)
- XXXVIII (=38) is lexicographically the last string which represents a valid Roman numeral
- 38 is the largest even number which cannot be written as the sum of two odd composite numbers
- 38 is the largest known even number which can be represented as a sum of two distinct primes in only one way (38 = 31 + 7)
- 38 is the sum of squares of the first three primes (2^2 + 3^2 + 5^2)

## Common Properties of 38

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.

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

The next *Ulam* number is uniquely the sum of two earlier distinct Ulam numbers.