Links
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http://www.mersenne.org/prime.htm
The Great Internet Mersenne Prime Search or GIMPS. This project brings together thousands of enthusiasts from around the world, who donate spare time on their computers to checking potential Mersenne primes. This sort of problem used to be undertaken by supercomputers, but modern desktop machines are almost as quick as the monster Crays of a decade ago.
There are several other such ‘distributed computing’ projects, many of which would welcome contributions from schools. See http://www.distributed.net/ for examples. A project that aims to use spare computer time to search for regular signals from radio telescopes, with a view to identifying intelligent life, is at http://setiathome.ssl.berkeley.edu/.
http://www.nrich.maths.org.uk/mathsf/journalf/may97/prob4.html
Sums of consecutive numbers.
http://www.nrich.maths.org.uk/
Many other investigations and problems at the Mathematics Enrichment online maths club.
