# Questions tagged [transcendence]

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### Exponential of Liouville Numbers

By Mahler classification of Transcendental real numbers (into the sets of $S$-, $T$- and $U$-numbers), we know that Any Liouville number is a $U$-number. $\log \alpha$ is either an $S$- or a $T$-...
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### Ground state energy of anharmonic oscillator: algebraic or transcendental?

Consider the quantum anharmonic oscillator, with Hamiltonian $H=p^2/2+q^2/2+gq^4$ for some real $g\geq 0$, with $p$ and $q$ obeying the usual Heisenberg commutation relations. For $g=0$, the ground ...
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### Reference request for results that involve the transcendence degree

Currently I'm reading "On the Decidability of the Real Exponential Field" by Macintyre and Wilkie and the Proof of Theorem 1.1 (page 462-464) uses two algebraic results that involve the ...
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### $a^b=b^a$ and algebraicity [closed]

Suppose $a$ and $b$ are reals such that $a^b=b^a$. If $a$ is algebraic, is $b$ algebraic too?
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### $f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?

Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied: (1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$. (2) ...
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### Number defined by a recursive binary sequence

In a math column in Scientific American many years ago, I encountered a peculiar binary sequence I describe below. Unfortunately I can't find a reference on this, so I would be grateful for any ...
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### Preservation of algebraically dependence for derivative [closed]

It is well-known (see Allouche monography for example) that if $f$ is an algebraic function over $K(X)$ then $f'$ is also algebraic. I wonder whether $f$ and $g$ are algebraically dependent, then $f'$ ...
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### algebraic independence of exponential functions

Let $a_1,\ldots,a_n$ be $\mathbb Q$-linearly independant algebraic numbers. Are the functions $e^{az},\ldots,e^{a_nz}$ algebraically independent functions (over $\mathbb C(z)$ or $\mathbb Q(z)$)? I ...
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### Number of generators of ideal if quotient field has certain transcendence degree

I am trying to prove a result which is used in A. Macintyre and A. J. Wilkie (1995), 'On the decidability of the real exponential field', in Odifreddi, P.G., Kreisel 70th Birthday Volume, CLSI, p. ...
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### Transcendence of solutions of $\sum_{i=1}^n a_i b_i^x = 1$

I am currently doing things related to the Akra–Bazzi theorem. One element in that theorem is the following: For $n>0$ and sequences of real numbers $a_i, b_i$ of length $n$, where all $a_i>0$ ...
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### Transcendence of products of certain real algebraic numbers

Let $$z := \prod_p p^{1/p^2},$$ where the product is over all prime numbers $p$, and we always take the positive real root. Is $z$ transcendental or algebraic, or (as I ...
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### Hermite Lindemann and transcendental reals

This is about compass and straightedge constructions, although I suspect nothing changes if we expand to real numbers algebraic over the rationals. Anyway, let $E$ be the "constructible numbers," ...
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Suppose I have two lists $\alpha_1,\dots,\alpha_k$ and $\beta_1,\dots,\beta_k$ of real numbers such that all $2k$ numbers are mutually algebraically-independent over the rationals. For each $i \in \{1,... 22 votes 3 answers 1k views ### Numbers with known finite irrationality measure greater than 2 For a real number$\alpha$, let the irrationality measure$\mu(\alpha) \in \mathbb{R}\cup \{\infty\}$be defined as the supremum of all real numbers$\mu$such that$$\left| \alpha-\frac{p}{q}\right|... • 337 -2 votes 1 answer 431 views ### Random Sequence : Definition of [closed] "A sequence of bits is random if there exists no Program shorter than it which can produce the same sequence." ~ Kolmogorov Q: How do the digits of Pi fall as a random sequence based on the above ... • 17 53 votes 2 answers 16k views ### Why is it hard to prove that the Euler Mascheroni constant is irrational? Philosophically why should proving that$\gamma$is irrational (let alone transcendental) be so much harder than proving$\pi$or$e$are irrational? • 1,513 5 votes 2 answers 2k views ### What was Lambert's solution to$x^m+x=q$? I've been thinking about Lambert's Trinomial Equation quite a bit, and I want to see his solution. The only solution I could find was in Euler's form, and I still don't quite understand how he got ... • 387 3 votes 0 answers 510 views ### quasi periodic continued fractions and powers of e, tanh, tan It is well known that some transcendental numbers like e.g. rational multiples of$e^{2/n}$with$n\in\mathbb N $, when written as regular continued fractions (R.C.F.), yield what can be called a ... • 13.2k 4 votes 0 answers 305 views ### Transcendental numbers that are "suspected" to be algebraically dependent without conjectured relation? I am experimenting with a solver for finding algebraic dependencies and would like to test it on more data sets. Are there transcendental numbers that are "suspected" to be algebraically dependent ... • 24.2k 24 votes 1 answer 2k views ### Is this seemingly novel irrational constant also transcendental? I recently discovered a constant that is constructed as follows:$\chi=\sum_{n=1}^\infty (\frac{\cos{n}}{2|\cos{n}|}+\frac{1}{2}) 2^{-n}$Furthermore I can prove that it is an irrational number ... 4 votes 1 answer 397 views ### Finding purely transcendental parts of field extensions If we have a field$K$such that$K\cong K(t)$(i.e. it is isomorphic to the field you get if you adjoin one transcendental) then is there necessarily a subfield$L\lt K$such that$L\ncong L(t)$and$...
An entire function $f$ is said to be of exponential type if there exist constants $c$ and $k$ such that $|f(z)|\leq c e^{k |z|}$. A famous result of Polya says if $f$ is an entire function of ...