The most celebrated prime number theorem is probably Fermat's Last Theorem which states there are no positive integer values $x$, $y$ and $z$ which satisfy $x^n+y^n=z^n$ for $n\ge 3$.
For $n=2$ we can find an infinite number of Pythagorean triples: $3^2+4^2=5^2$ (meaning $9+16=25$) being the most familiar.
In 1637 French mathematician Pierre de Fermat conjectured that no such integer values could be found for larger values of $n$. In fact he proved it for the case $x^4+y^4=z^4$. Fermat claimed to have a general proof of his conjecture but none was never found - even following a careful search of his papers after his death. Fermat's last theorem remained unproved until 1995 when British mathematician Sir Andrew Wiles published a proof using mathematics unknown to Fermat.
The prestigious journal Nature is carrying a story today connected to the twin prime conjecture. Mathematicians have shown that infinite pairs of primes exist. The difference is that they've proved that an infinite number pairs exist spaced by not more than 70 million. This is somewhat larger than prime pairs spaced by a single even number. But it represents significant progress on one of the hardest, open problems in mathematics.