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When someone asks a question asking to prove a negative (such as "is this true" or the like), should it be left up forever, or should it be closed after a certain amount of time, and would an answer of "there is no such mention in Bavli or Yerushalmi, or Medresh Rabba or Tanchuma" (which I've been tempted to give several times) appropriate?

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It depends on the subject and relevant body of source material, but generally, something like "I've studied this subject extensively, including all of the relevant sections in the Talmud, etc., and have never come across such a thing." can be a valid answer. The more basis you include to back up your assertion, the stronger your answer is.

For some examples of answers roughly of this form, see: Have Jews believed that Ezra was the son of God?

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