Using software from the GIMPS project, Cooper and Steven Boone found the 43rd known Mersenne prime on their 700 PCcluster on December 15, 2005. The prime, 230402457 − 1, is 9,152,052 digits long and is the ninth Mersenne prime for GIMPS.
Cooper and Boone became the first GIMPS contributors to find two primes when they also found the 44th known Mersenne prime, 232582657 − 1, which has 9,808,358 digits (or M32582657 for short). This prime was discovered on September 4, 2006 using a PC cluster of over 850 machines. This is the tenth Mersenne prime for GIMPS.
On January 25, 2013, Cooper found his third Mersenne prime, which is 257,885,161 − 1. It is also the largest known prime number as of January 2014.
Cooper's own work has mainly been in elementary number theory, especially work related to digital representations of numbers. He collaborated extensively with Robert E. Kennedy. They have worked with Niven numbers, among other results, showing that no 21 consecutive integers can all be Niven numbers, and introduced the notion of tau numbers, numbers whose total number of divisors are itself a divisor of the number. Independent of Kennedy, Cooper has also done work about generalizations of geometric series, and their application to probability.