There is no Constitutional requirement for a candidate to produce a birth certificate. You have to be a natural born citizen, whatever the Congress and the Supreme Court says that is (before the 14th Amendment, you had to be white to be a citizen, though Congress could just proclaim that you were white, as it was known to have done). There is no constitutionally set legal requirements on proving that you are. You should be at least 35 years old. The constitution demands that you have lived in the country for at least 14 years (not necessarily in one chunk of time). Lastly, you much take the oath of office (which. by the way, doesn't mention God.)