# Questions tagged [transcendental-number-theory]

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### Characteristics of numbers that satisfy inequality related to diophantine approximation with perfect squares

This question has come up in my algorithms and physics research. I apologize if this is very basic, but I am new to number theory and it seems this is a number-theoretic question. What can we say ...
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### Is there any analogue of $\pi$ for non-Archimedean Euclidean fields?

Let us recall that a Euclidean field is an ordered field in which every positive element has a square root. Given a Euclidean field $F$, we can consider the Euclidean'' plane $F^2$ endowed with the ...
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### "Interlacement" of transcendental numbers

Let us consider two transcendental numbers whose decimal representation is $$0.a_1a_2a_3a_4...$$ $$0.b_1b_2b_3b_4...$$ Is $$0.a_1b_1a_2b_2a_3b_3a_4b_4...$$ also a transcendental number?
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### Are there rational $a,b$ with $a+be=1/\ln 2$?

Are there rational $a$ and $b$ with $$a+be = \frac{1}{\ln 2}\ ?$$ I used the absence of rational solutions repeatedly in this answer. Here is a proof using Schanuel's conjecture: $e^q$ is ...
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### Why is it easier to prove $e$ is transcendental than $\pi$?

Why is it easier to prove $e$ is transcendental than $\pi$? I noticed that the proofs of $\pi$'s transcendence are much longer and have more details to check than those of $e.$ My guess is that it's ...
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### Transcendence of values of Fredholm series at algebraic arguments

Let $d$ be an integer greater than $1$ and let $f(z)=\sum_{n\ge0}z^{d^n}$ be the Fredholm series. It is well-known that the Roth-Ridout theorem implies that $f(r)$ is a transcendental number for all ...
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### Does logarithms conjecture imply Gelfond conjecture?

Assume that logarithms conjecture is true: Let $a_1,\cdots,a_n$ be algebraic numbers such that $\log(a_1),\cdots,\log(a_n)$ are $\mathbb Q$-linearly independant. Then, $\log(a_1),\cdots,\log(a_n)$ are ...
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### Testing whether $e^x+ax^2+bx+c$ has a zero

What is the simple test with exponential polynomials to determine whether $$f(x)=e^x+ax^2+bx+c$$ has a positive zero? This was prompted by the question about discriminants here. We have an ineffective ...
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### How to prove that the solution to $x^{x+1}=(x+1)^{x}$ is transcendental?

I was asked by an high school student if there is an algebraic way to find the exact value of the solution to the equation $$\label{eq} x^{x+1}=(x+1)^x$$ Let us define that ...
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### On $\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}$ and $\sum_{n=1}^\infty\frac1{p_1\cdots p_n}$

For $n=1,2,3,\ldots$ let $p_n$ denote the $n$-th prime number. Question. Are the two numbers c_1=\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}\ \ \ \ \text{and}\ \ \ \ c_2=\sum_{n=1}^\infty\frac1{p_1\...
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1 vote
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### Is the “amalgamation” of an enumerated infinite collection of absolutely normal real numbers always absolutely normal?

Let $S$ denote an enumerated infinite collection of absolutely normal (i.e. normal in all integer bases greater than or equal to $2$) real numbers (with no repetitions) such that any natural number ...
555 views

### Do we have an algorithm for comparing $e^e$ with rationals?

Do we have an algorithm for comparing $e^e$ with rationals, with a known time to convergence? In a non-constructive sense, there obviously is an algorithm. If $e^e$ is some rational $q_0$, then we ...
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### Picture of Lambert's proof that $\pi$ is irrational?

With a suitably generous notion of "picture proof" or "proof without words" or "geometric proof," there do exist such proofs of the irrationality of square roots and even ...
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