I purchased LR some years ago & want to download it again on my new computer

If you registered with Adobe (or purchased from Adobe) then they will have your serial number at account.adobe.com, sign in with the e-mail you used when you made the purchase, then click on View Billing History, then click on Products, you will see your serial numbers. Again, this assumes the IF at the start of that sentence is correct. When I do this, I see my Lr 2 serial number (and Lr 3 and Lr 4 and Lr 5 and Lr 6 serial numbers).

If you don’t see your serial number there, then the only other way to retrieve your serial number is if the computer that you used to run Lr is still available.

I agree with ​@Jim Wilde that there is little chance that you will find a download file for Lr 1 or Lr 2, and virtually zero chance that it will run on Tahoe.

ADVICE: from now on, no exceptions, you need to mark down all of your serial numbers (not just Lr) in a text file that is shared to the cloud, so you can always find your serial numbers if you need them.

What are you looking to download, the serial number or the Lightroom application itself? If it’s just the serial number have you checked your Adobe Account online to see if you registered the serial number there?

If it’s a download of the Lightroom application that you’re looking for, using a serial number means that the version of Lightroom that you are looking for is between Lightroom 1 and 6 (after LR6 the application name became Lightroom Classic in 2017 and can only be used with a subscription). The chances of you being able to locate a download of the correct version, and then successfully install it on a current MacOS operating system are almost non-existent. Can you recall which version of Lightroom you purchased, and can you tell us what operating system version you are running on your MBP.