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.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
You can see a longer list of prime numbers if you like.
You can explore the Goldbach conjecture interactively with the Prime Machine applet.
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.
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:
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.
Fine print, your comments, more links, Peter Alfeld, PA1UM