Plots of Various Number Theory Functions

function

number of divisors of n

maximum value

500

This Demonstration plots various number theory functions.

A perfect number

n

has the property that

2n

equals the sum of the divisors of

n

. (For example, 6 is perfect because

2×6=1+2+3+6

.) It is not known if there are infinitely many perfect numbers.

Every positive integer can be expressed as the sum of four squares. The numbers to be squared can be positive, negative, or zero and we count different orderings of the summands as different sums. There are some positive integers that cannot be expressed as the sum of three squares; those that can are of the form

r

4

(8s+7)

.

Goldbach conjectured that every even number is the sum of two primes.