Hi Don,
Can you clarify what you mean by product key? Adobe Creative Suite products utilize 24 digit serial numbers for installation. If you purchased your software through a third-party reseller, you may have originally received a redemption code, which is a string of alpha-numeric characters. However, the redemption code is converted to a serial number when you first input it. For more information on redemption codes, please see this document: Redemption code help.
If you are trying to retrieve the serial number from your other computers through System Info or similar means, some digits may not be displayed.
Since it sounds like you were not able to find the number by going through the steps in the document I linked earlier, you may want to Contact Customer Care. Click the blue "Still Need Help" option at the bottom of the Contact Customer Care screen and select chat. Although chat is not able to provide technical support for older software, they can clarify which licenses and serial numbers are associated with your account. They can also search phone numbers and e-mail addresses if you feel there is a serial number missing from your account.
Let us know how it goes.
Del