Pressed for a precise definition, many mathematicians might have difficulty saying exactly what a proof is. In mathematical practice, a proof is any sufficiently detailed convincing mathematical argument that logically establishes the conclusion of a theorem from its premises. Successful proofs of new theorems often succeed by introducing new mathematical ideas or methods, and one gains mathematical insight from the proof beyond merely learning the truth of the theorem itself. We value such proofs because we might use the new methods to answer other questions that intrigue us. To prove a
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