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Any other definition for algebraic number than the root of algebraic equation?

Given a polynomial $p(x)\in \mathbb{Q}[x]$, it is known that its roots can be obtained in terms of the coefficients of the polynomial by formulas involving the usual algebraic operations (addition, su …

Suppose A is Diophantine Equations which have a variable x,such that every other variable has function of x the variable as it's solution.For example,a Diophantine equation in three variables has …

Any grammar for the language $$L =a^p,\text{ $p$ is prime and }p\in \mathbb{N}?$$ Is such a grammar related to any question of number theory like RH or the conjecture of twin primes?

For S ,any given c.e.set,does there exist a M (integer) and a partially computable function outputing every element of S the c.e.set ,such that $\forall x\in S,\exists n x=f(n)$ and $x=f(n)\leq …

Suppose $S$ is set of numbers such that every number in it expands in decimal digits,every digit is 0 or 1,and $\lim_{n\rightarrow\infty}\frac{C_{n}(0)}{n}=\frac{1}{2}$ where ${C_{n}(0)}$ and ${C_{n} …

I want to know what the critical idea behind Hardy-Littlewood circle method is. It seems that they divide the circle into major arcs and minor arcs to ignore the singularities of generating function t …

Any closed form for series like $$F(x)=\sum_{p=2}^{\infty}x^p,\quad p\text{ is prime}$$ or $$F(x)=\sum_{i=0}^{\infty}x^{i!}\quad ?$$ More generally, we can obtain a power series from decimal expansion …

Let $$f(x)= \sum_{n=1}^{\infty}a_n x^n$$ where $a_n\in \{0,1\}$, and the $f(x)$ has a natural boundary. By the way, $$f(x)= \sum_{n=1}^{\infty}a_n x^n$$ is rational function or transcendental one on $ …

The continued fraction $$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance, is known explicitly as a ratio of Bessel function values and is (I believe - SS) known to be transcen …

Above the Comments in the article continued fraction of algebraic numbers, there are some words on the unboundedness/cycle of coefficients of continued fraction of algebraic numbers "Thus, contrary t …

Roughly speaking,Kolmogorov Complexity of a bits string or a description is the minimal length of programs outputing a bits string,and height of rational number is logarithm of the largest numerator o …

What is the probability that a randomly chosen number from the set of c.e.numbers is period(number)? What is the probability that a randomly chosen number from the set of computable numbers is pe …

Given a partially computable function, is there an analytic complex function which is equal to it at every point of it's domain? Or under what condition does a partially computable function correspond …

I vaguely recall that resolution of singularity may be linked to continued fracton, possibly it is cusp that links to CF. Could any one give concrete reference and give example? Thanks.