I found my papers after (a) five minutes' search.
We don't need 'a' in this sentence, do we?
If I insert an attribute, does 'a' become a must?
I found my papers after a confused five minutes' search.
I spoke to her during dinner.
I spoke to her during a formal dinner.
You'll notice that as soon as you use an adjective to describe 'dinner' you need an article.
Hope that helps