First if a prime is found the server notifies the project leaders...
They go ahead an contact the user almost imediately, the first thing they do is jump up and down several times and call their friends
Then they probably simultaneously recheck the k/n pair with the sob program themselves and contact the user.
Of course our client doesn't prove a number is prime, it proves a k/n pair is not prime with a very high degree of probablility.
Once they are confidant, at least a day, they try to prove the k/n pair prime.
To actually prove a particular k/n pair is prime takes weeks of time supercomputer time, also different, computers, programs, and O/S, are also used to verify.
Hope this helps.