The Goldbach conjecture states that all positive even integers greater than or equal to four can be expressed as the sum of two odd prime numbers.

Cristian Goldbach (1690-1764) submitted his conjecture to Euler who dismissed it as trivial, however as of today (2003) it has not yet been proved.

See this page for more details.

"Upon looking at these numbers, one has the feeling of being in the presence of the inexplicable secrets of creation."
D. Zagier from "The first 50 million prime numbers"

143-Year-Old Problem Still Has Mathematicians Guessing
The Riemann hypothesis, first tossed off by Bernhard Riemann in 1859 in a paper about the distribution of prime numbers, is still widely considered to be one of the greatest unsolved problems in mathematics.