Ivan Niven wrote a celebrated elementary proof of the Irrationality of pi. His one-page 1947 article turns a hypothetical rational expression for into an impossible integer integral.[ref]
A simple proof that pi is irrational by Ivan Niven. Original byline and complete proof, p. 509
A factorial-normalized polynomial has integer-valued endpoint derivatives under the assumption . Repeated integration by parts makes its integral against an integer. Yet positivity and the factorial denominator make that same integral lie strictly between zero and one for sufficiently large degree. The contradiction demonstrates how carefully designed analytic estimates can establish an arithmetic fact.[ref]
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