Without using numbers and using only permitted math operators, write an equation using the constants $e$, $i$, $\pi$ and $\phi$. Allowed operators..exponent...multiplication..subtraction...only. Use the constants minimal number of times.


closed as too broad by noedne, PiIsNot3, Mr Pie, Glorfindel, Jeff Zeitlin Apr 30 at 17:47

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    $\begingroup$ There are many answers; I believe this is too broad. $\endgroup$ – noedne Apr 30 at 16:14
  • $\begingroup$ Excluding trivial...answer will be less than 10 characters $\endgroup$ – Uvc Apr 30 at 16:34
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    $\begingroup$ Hi @Uvc, welcome to Puzzling SE! (Take the tour if you haven't already!) Unfortunately, as your question stands, it is too broad and thus will likely be closed. Please see here for information about what is considered on-topic here. Thanks! $\endgroup$ – PiIsNot3 Apr 30 at 16:53
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    $\begingroup$ Can you clarify as to what a "character" and "math operator" is? $\endgroup$ – PiIsNot3 Apr 30 at 17:35
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    $\begingroup$ What counts as "nontrivial?" $\endgroup$ – noedne Apr 30 at 17:41

How about this

$e^{i \pi} = \phi - (\phi \times \phi)$

  • $\begingroup$ Clever! Wish I didn't reach my daily voting limit! $\endgroup$ – Mr Pie Apr 30 at 16:48
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    $\begingroup$ Why don't use ROT13(artngvir cuv bire cuv) on RHS? $\endgroup$ – athin Apr 30 at 17:04
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    $\begingroup$ @athin indeed, that would use less symbols, but the original answer above showcases the beauty behind each of the values via their delicate properties (e.g. like how $\phi^2-\phi-1=0$ and the obvious LHS). But eh, aesthetics over poetics! :D $\endgroup$ – Mr Pie Apr 30 at 17:23
  • $\begingroup$ Thanks Mr.Pie...every math book I have seen so far mention famous Euler equation but I have not seen it tied to the unique quadratic equation involving the golden ratio.. $\endgroup$ – Uvc Apr 30 at 17:31
  • $\begingroup$ @Uvc it is simply a matter of substitution, as a matter of fact. I mean, given that $\phi$ is the root of a quadratic equation, it is paired with another root. Since this quadratic is the minimal polynomial, it follows that $\phi$ and its conjugate $1-\phi$ are the two distinct roots, respectively; so I could even tie Euler's identity with the latter root instead, if I pleased, but we are not given that in the actual puzzle itself :) $\endgroup$ – Mr Pie Apr 30 at 18:02

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