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I am trying to prove that $\pi+\pi^2$ is irrational assuming that $\pi$ is transcendental.

My work: I noticed that at least one of $\pi+\pi^2$ and $\pi-\pi^2$ is transcendental. (Because algebraic numbers are closed under addition.) However, similar methods probably cannot resolve the question, because otherwise they could be used to show that $\pi+e$ is irrational which is an open problem. This makes me think that a solution must use the fact that the numbers $\pi$ and $\pi^2$ are related.

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    $\begingroup$ Hint: quadratic equation. $\endgroup$
    – Anne Bauval
    May 16 at 12:39
  • $\begingroup$ What do you think transcendental means? $\endgroup$
    – Thomas Andrews
    May 16 at 12:50
  • $\begingroup$ Not a solution of a polynomial equation with rational coefficients? (I know it was pretty stupid of me, but now I get it.) $\endgroup$
    – Adam
    May 16 at 15:05

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If $\pi+\pi^2$ were rational then $\pi$ would be algebraic.

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