I have $3$ numbers which seem to be transcendental, but probably very difficult to be proven transcendental. PARI/GP gives no indication that one of the numbers is algebraic. The first one $$\int_0^1 \ln(x^2+1) dx=\ln(2)+\frac{\pi}{2}-2$$ This number has the same numerical status as $\ln(2)+\frac{\pi}{2}$, which is the sum of two transcendental numbers. Of course, this does […]

Regaridng to the question Real numbers which are writable as a differences of two transcendental numbers now, it is important to know answer of the following question: Is it true that for every real number $x\neq 0$ there exist linearly independent transcendental numbers $\alpha$ and $\beta$ (i.e., $r\alpha+s\beta=0$ implies $r=s=0$, for every rational numbers $r,s$) […]

it seems at least superficially plausible that for real $x \ne 0$ then at least one of $\{x, \cos\;x\}$ is transcendental over $\mathbb{Q}$. has this assertion been proved to be true or false?

A complex number that has transcendental real part is always transcendental? How about in the case of imaginary part?

I need to prove that the set of all transcendental numbers is dense in $\mathbb{R}$, and to that end, I have written the following proof: Let $\mathbb{T}$ denote the set of transcendental numbers in $\mathbb{R}$. First, notice that $\mathbb{T} \subset \mathbb{R}\setminus \mathbb{A}$, where $\mathbb{A}$ denotes the algebraic numbers in $\mathbb{R}$. Next, since $\mathbb{A}$ is a […]

Are there any transcendentals whose real or imaginary components have not been found in exact form?

It is often said that $\pi$ contains any string of digits. But does the property “transcendental” imply “contains any string of digits?

Weierstrass proved the result [Lindemann-Weierstrass theorem] that if $a_1, \cdots, a_n$ are reals linearly independent over the rationals, then $e^{a_1}, \cdots, e^{a_n}$ are algebraically independent. I would like to know if the result holds for infinitely many numbers. Explicitely, if $\{a_1, a_2, \cdots \}$ is an infinite family of real numbers such that every finite […]

I don’t really know a lot about this subject but I was wondering if the product of a transcendental number by an integer is transcendental?

Is it possible to add or multiply (no subtraction) only positive transcendental numbers and yield a solution that is algebraic? Exponential manipulation is excluded from this question, as $e^{\ln2} = 2$ EDIT: Also, excluding reciprocals of other transcendental numbers and subtraction?

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