What is known about transcendence of the numbers of the form ${\pi}^{r}$ where $r \in \mathbb Q \setminus \{0,1\}$? For which $r$ are they proven to be transcendental?
2026-03-28 17:40:14.1774719614
Transcendence of the rational powers of $\pi$?
135 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in REFERENCE-REQUEST
- Best book to study Lie group theory
- Alternative definition for characteristic foliation of a surface
- Transition from theory of PDEs to applied analysis and industrial problems and models with PDEs
- Random variables in integrals, how to analyze?
- Abstract Algebra Preparation
- Definition of matrix valued smooth function
- CLT for Martingales
- Almost locality of cubic spline interpolation
- Identify sequences from OEIS or the literature, or find examples of odd integers $n\geq 1$ satisfying these equations related to odd perfect numbers
- property of Lebesgue measure involving small intervals
Related Questions in PI
- Two minor questions about a transcendental number over $\Bbb Q$
- identity for finding value of $\pi$
- Extension of field, $\Bbb{R}(i \pi) = \Bbb{C} $
- ls $\sqrt{2}+\sqrt{3}$ the only sum of two irrational which close to $\pi$?
- Is it possible to express $\pi$ as $a^b$ for $a$ and $b$ non-transcendental numbers?
- Is there an essential difference between Cartwright's and Niven's proofs of the irrationality of $\pi$?
- How and where can I calculate $\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots\right)\left(1+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+\cdots\right)?$
- Is $\frac{5\pi}{6}$ a transcendental or an algebraic number?
- Calculating the value of $\pi$
- Solve for $x, \ \frac{\pi}{5\sqrt{x + 2}} = \frac 12\sum_{i=0}^\infty\frac{(i!)^2}{x^{2i + 1}(2i + 1)!}$
Related Questions in TRANSCENDENTAL-NUMBERS
- Two minor questions about a transcendental number over $\Bbb Q$
- Is it possible to express $\pi$ as $a^b$ for $a$ and $b$ non-transcendental numbers?
- Is it true that evaluating a polynomial with integer coefficients at $e$, uniquely defines it?
- Is $\frac{5\pi}{6}$ a transcendental or an algebraic number?
- Is there any intermediate fields between these two fields？
- Is there any pair of positive integers $ (x,n)$ for which :$e^{{e}^{{e}^{\cdots x}}}=2^{n}$?
- Why is :$\displaystyle {e}^\sqrt{2}$ is known to be transcedental number but ${\sqrt{2}}^ {e}$ is not known?
- Irrationality of $\int_{-a}^ax^nn^xd x$
- Proving that $ 7<\frac{5\phi e}{\pi}< 7.0000689$ where $\phi$ is the Golden Ratio
- Transcendence of algebraic numbers with Transcendental power
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
All of them.
If $\theta=\pi^{a/b}$ is algebraic then so is $\theta^b = \pi^a$ and so is $\pi$.