 ## Some Simple Unsolved Problems

One of the things that turned me on to math were some simple sounding but unsolved problems that were easy for a high school student to understand. This page lists some of them.

## Prime Number Problems

To understand them you need to understand the concept of a prime number. A prime number is a natural number greater than 1 that can be divided evenly only by 1 and itself. Thus the first few prime numbers are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...

You can see a longer list of prime numbers if you like.

### The Goldbach Conjecture.

Named after the number theorist Christian Goldbach (1690-1764). The problem: is it possible to write every even number greater than 2 as the sum of two primes? The conjecture says "yes", but nobody knows.

You can explore the Goldbach conjecture interactively with the Prime Machine applet.

### Prime Twins

A prime twin is a pair of primes that differ by 2. Examples for prime twins are: (3,5), (11,13), and (1000000007,1000000009). The largest known prime twin is Each member of this twin comprises 11,713 decimal digits! How many prime twins are there? Are there infinitely many or finitely many. In the latter case, how many? Nobody knows the answers to these questions.

You can also explore prime twins interactively with the Prime Machine applet.

## Perfect Numbers

A perfect number is one which equals the sum of its proper divisors. If the sum exceeds the number it is abundant, otherwise it is deficient. For example: the number 18 is abundant since

18 is less than 1 + 2 +3 + 6 + 9 =21,

the number 15 is deficient since 15 is greater than 1+3+5 =9 and

6 =1+ 2 + 3 is perfect.

Perfect numbers have been studied since antiquity. It is known that all even perfect numbers are of the form where both p and (*)

are prime. Primes of that form are called Mersenne Primes. It can be shown that p must be prime for (*) to be prime. As of December 2002, 39 Mersenne Primes are known. There are thus 39 known even perfect numbers. The following questions have not been answered to date:

• How many even perfect numbers are there? More precisely, are there infinitely many, or finitely many? In the latter case, just how many are there?
• Are there any odd perfect numbers?

## Solved Problems

The Four Color Map Problem and Fermat's Last Theorem were famous open problems when I went to High School, but they have since been solved.

[06-Dec-1999]