query: Could $\pi$, raised to some power can give rational result?
positive_passages.docid: 143054
positive_passages.text: Can you raise $\pi$ to a real power to make it rational?
negative_passages.docid: 61794
negative_passages.text: Interesting and unexpected applications of $\pi$
negative_passages.docid: 43659
negative_passages.text: Why is $\pi$ so close to $3$?
negative_passages.docid: 10882
negative_passages.text: irrationality of $\sqrt{2}^{\sqrt{2}}$.
negative_passages.docid: 97071
negative_passages.text: Is the cube root of a prime number rational?
negative_passages.docid: 30523
negative_passages.text: A binary irrational with bits defined by primes
negative_passages.docid: 44400
negative_passages.text: $e^{2\pi i x} = (e^{2\pi i})^x$: What happens if x is rational?
negative_passages.docid: 87596
negative_passages.text: Is $\sum_{p}\frac{1}{p^{2}}$ irrational?
negative_passages.docid: 56464
negative_passages.text: Proving the irrationality of $e^n$
negative_passages.docid: 132928
negative_passages.text: Digits of $\pi$ forming primes?
negative_passages.docid: 24706
negative_passages.text: Is there a rational number between any two irrationals?
negative_passages.docid: 155112
negative_passages.text: Are there more rational numbers than integers?
negative_passages.docid: 81957
negative_passages.text: How many roots have a complex number with irrational exponent?
negative_passages.docid: 103376
negative_passages.text: Is $\sin(p \pi/q)$ necessarily an algebraic number?
negative_passages.docid: 7500
negative_passages.text: $\arctan$ of a square root as a rational multiple of $\pi$
negative_passages.docid: 4946
negative_passages.text: Non-existence of irrational numbers?
negative_passages.docid: 44600
negative_passages.text: How can I prove: if $p $ is prime and $n>1$, then $ p^{\frac1n} $ is irrational?