No, you don't. Just type in your registration number on the section on the N-400 where that information is asked. If you feel so worried about it, then on the day of the interview bring the printout of your SS registration, which you can print from the SSR site.
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It is not a requirement and they usually will not ask for it. However, if they ask for it in a Request For Evidence (RFE) later for some reason then you need to submit it.