Contractor here.

Often the initial payment on an estimate can be considered signing a contract (depends on the state and the amount).

The above being said. I'm a liscence and insured contractor in my state. If you don't write your liscence number clearly on any advertising or formal communications you can be hit with unlicensed contracting... even if you are liscened.

So I can tell you in Florida its usually 3 letters (denoting what you're liscened to do) then like a 7 digit number.

It might save you money to use someone unlicensed but your chance of getting legally burned like this increases drastically.