# 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 773

- 773 is the smallest prime p with property that appending or prepending any single decimal digit to p does not produce a prime
- 773 is the smallest home prime that is home to five primes
- 773 is the smallest Henry VIII prime: it is left-truncatable, but can't be left-truncated from another prime
- 773 is the smallest odd number n so that n+2
^{k} is composite for all k less than n
- 773 is the largest three-digit Unholey prime: such primes do not have holes in their digits

## Common Properties of 773

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.

The number n is *evil* if it has an even number of 1's in its binary expansion.

Guess what odious numbers are.

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.

A *prime* is a positive integer greater than 1 that is divisible by no positive integers other than 1 and itself.

Prime numbers are opposite to composite numbers.

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