There are many interesting discoveries made by mathematical distributed computing projects. 

* The Great Internet Mersenne Prime Search: http://www.mersenne.org/
* ABC@Home: http://en.wikipedia.org/wiki/ABC@Home
* PrimeGrid: http://en.wikipedia.org/wiki/PrimeGrid
* Ramsey@Home: http://boinc.berkeley.edu/wiki/Ramsey@Home

Their discoveries don't have an impact in the same way that theorems do, but from time to time resolving a theorem boils down to computation, and most of the participants are probably interested amateurs.