When you purchase QuickBooks through the Intuit web store, Intuit gives you an installer file to download, a license number, and a product number. The license number and product number are available on the web store for three years, after which Intuit removes the purchase record.
I wanted to move my QuickBooks installation to a new computer, but I had no record of my license number and product number. The solution? Run regedit and view registry key HKEY_LOCAL_MACHINE/SOFTWARE/Intuit/QuickBooksRegistration/15.0/pro. (I have a license for the "pro" version of QuickBooks.) Look for entries with names like InstallID and LicenseNumber. LicenseNumber is your license number, and InstallID is your product ID. I recovered my license and product numbers, installed QuickBooks on the new PC, and I was all set.