# 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 1681

- 1681 (= 41
^{2}), is smallest composite number generated by the Euler polynomial n^{2} + n + 41
- 1681 is the only four-digit square not ending in two zeroes such that each of its two 2-digit parts is a square
- 1681 is the smallest number n such that n*prime(n)+(n+1)*prime(n+1) is a perfect square

## Rare Properties of 1681

The number n is a *square* if it is the square of an integer.

## Common Properties of 1681

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 *odd* if it is not divisible by 2.

Numbers that are not odd are even. 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, p^{2} also divides n.

How much power? They all can be written as a^{2} b^{3}.