Hi Jim/(Anyone who lost their serial number),
When you say you have purchased Download Order, you need to first confirm whether you have purchased the product via Adobe online OR from a reseller? Few resellers do provide download links too. If you have purchased the product on Adobe online then,
1. Confirm the email address which you have used to place the Order. (You may have sometimes used your Old/different email address to place the Order)
2. Confirm whether you have purchased a Retail version Or a Volume Licensing Orders.
If you have purchased Retail version over adobe online,:
a. Then login to adobe.com and navigate to "My Adobe" > "My Orders". You will see the Order dates. Click on the date on which you have placed the Order and there you will find the serial number as well as the Download link.
b. If you have registered it under your Adobe account, then navigate to "My Adobe" > "My Products".
If it is a Volume Licensing Order:
a. Navigate to licensing.adobe.com, enter your credentials and click on "Licences" > Retrieve the serial number. Select the Deploy-to ID or End-User ID which is used to place the Order and click on Search. You should now see the serial numbers.
Lastly, if you have purchased the Download Order from a Reseller, you usually might have their Login account. Login to their account and you may find the serial number. You can also contact the reseller from where you have purchased it.
If these suggestions doesn't work, then contact Adobe Chat support/Phone who will help you on this. You may need to send the Proof of purchase if purchased from a Reseller..