I always give my (deceased) dad's old SSN. (You can find anyone's SSN in the Social Security Death Index, online.) Most medical offices will never catch it. What they want it for is if you never pay - they want to report it to a credit reporting agency like TransUnion. As long as their computer has a 9 digit number in it, it's happy. The office has no way of knowing what your real number is.
The form doesn't say YOUR Social Security Number; it just says SOCIAL SECURITY NUMBER !
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