What computer operating system for your Premiere Elements 13?
Are you sure that you have the serial number and not a redemption code or product code?
Are you doing the download and install with the antivirus and firewall(s) disabled?
If the above has not led to the solution, then you need to contact Adobe. Here we are user to user. We are not Adobe.
Please click on the following link to reach Adobe Chat
Adobe ID, Signing In
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If the link does not hold its set, then you will need to navigate to the Chat Panel using the above titles as guides.
Please let us know if you are OK with the above information.