# 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 71

- 71 is the only two-digit number n such that (n
^{n} − n!)/n is prime
- 71 is the largest of the supersingular primes, i.e., the set of primes that divide the group order of the Monster group
- 7! + 1 = 71^
^{n}. Moreover, 71 is the largest known number n such at its square is 1 plus a factorial
- 713 = 357911: the concatenation of five consecutive primes

## Rare Properties of 71

The n-th *Google* number is the first n-digit prime found in the decimal expansion of e.

They are named *Google* numbers because of the unusual hiring ad that *Google* put up.

## Common Properties of 71

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.

A prime number is called a *twin* prime if there exists another prime number differing from it by 2.