2

I tried for a few hours to come with a proof that $π^e$ is irrational. I mainly tried with the method "proof by contradiction" and didn't use calculus at all, but couldn't come up with a proof.

Can anyone give a proof that $π^e$ is irrational? I would appreaciate if it was without using calculus. because that's more fun, but it's completly fine using calculus too :) Just want to see a proof.

Thanks in advance!

Casimir Rönnlöf
  • 1,054
  • 6
    This is unknown – Peter Jan 19 '19 at 18:28
  • @Peter What? Seriously? Without calculus even? Ooo that's a fun challenge for a 14 year old math lover :DD – Casimir Rönnlöf Jan 19 '19 at 18:34
  • 4
    I'm voting to close this question as off-topic because it involves speculation about an open problem in mathematics. I don't think that such speculation is on-topic here (though I suspect that a question about what is known about the ir/rationality of $\pi^\mathrm{e}$ might be both (1) on-topic and (2) already answered here somewhere). – Xander Henderson Jan 19 '19 at 18:35
  • 2
    Good hazing for a young mathematician! – mathcounterexamples.net Jan 19 '19 at 18:36
  • Link to a list of combinations of $\pi$ and $\mathrm{e}$ which are not know to be transcendental (but might be): https://en.wikipedia.org/wiki/Transcendental_number#Possibly_transcendental_numbers – Eric Towers Jan 19 '19 at 18:36
  • Possibly relevant: https://math.stackexchange.com/questions/767240/is-large-pie-rational, https://math.stackexchange.com/questions/1952889/pi-and-e-problem – Xander Henderson Jan 19 '19 at 18:37
  • @EricTowers ty for that page – Casimir Rönnlöf Jan 19 '19 at 19:00
  • 1
    Note that $e^\pi$ is proven to be transcendental on the same wiki page. However, the proof is only easy if we can use the Gelfond-Schneider theorem and apply it to $(-1)^{-i}$. – Klaas van Aarsen Jan 19 '19 at 19:22

0 Answers0