You can't download a serial number. You can download installation files.
A serial number is printed on the cover of the installation CDs/DVDs if you have perpetual licenses. If you have registered your product you'll find the serial number under your Adobe account. Login with your Adobe ID on the Adobe's website.
Which version of Premiere Pro you're talking?
Adobe has removed older version from the public web sites.
You have to contact Adobe directly for a download link. Have your serial number and details of original purchase available. Make it clear that you doesn't need support for an older product but need a download link. Click the link below and scroll down the page to Individual “Sign in to contact us” It should be possible to start an on-line chat with an Adobe agent.