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Answer: Paul Erdős
More information: Erdős published at least 1,525 mathematical papers during his lifetime. He engaged more than 500 collaborators and devoted most of his time to mathematics up until his death in 1996 at the age of 83 years old.
Answer: Pierre-Simon Laplace has an Erdős number of at most 14!
More information: According to "The Erdös number project" a path to Laplace was found by Leonid Yanushevich giving the famous mathematician an Erdös number of at most 14.
More information: Of course, as time passes, the smallest Erdős number that can still be achieved will necessarily increase, as mathematicians with low Erdős numbers die and become unavailable for collaboration.