Certificate numbers can be found in a variety of documents in the public domain. Three common examples are below.
Certificate numbers are printed in red on all US Certificates of Naturalization or Certificates of Citizenship issued since September 27, 1906. (see example below).
Certificate numbers may also be found on some indexes to court naturalization records. Court indices primarily reference court petition and declaration documents. For that reason one must take care not to confuse the petition number with the certificate number. They are not the same. (see example below of federal court naturalization index cards from the National Archives).
Many naturalization petition forms filed after June 1906 contain a blank for the final naturalization certificate number. The certificate number may be noted in this blank on the back of the form. (see example below of a petition for naturalization found among federal court records at the National Archives).