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### Time-efficient way of calculating the least number of 1s in a representation of $n$ using only the operations $+,!$

This was inspired by the following paper: J. Arias de Reyna, J. van de Lune, "How many $1$s are needed?" revisited, arXiv link. It might help explain my question better, because my question is ...
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### An effective way to tell if the saturation of a homogeneous ideal is the irrelevant ideal

Let $\Bbbk$ be an algebraically closed field, let $R$ denote the graded ring $\Bbbk[x_0, \dotsc, x_N]$, and let $f_1, \dotsc, f_n \in R_m$ be nonconstant homogeneous polynomials. Then the common ...
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### Advances and difficulties in effective version of Thue-Roth-Siegel Theorem

A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result: If $\alpha$ is a real ...
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### Question related to Diophantine approximations and Roth's theorem

The following question came up in my arithmetic geometry course yesterday. Suppose $\alpha$ is an irrational real algebraic integer, and suppose $\epsilon >0$ is given. Then by Roth's theorem there ...
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Let $K$ be a number field, and $p$ be a rational prime. Then the Chebotarev Density Theorem implies we can find primes $v$ and $w$ of $K$ of degree 1 which are split and nonsplit respectively in $K[\... 0answers 844 views ### Effective upper bound on large prime gaps; or, what is the first prime after a googolplex? Question What is the best known effective upper bound on the prime gap following x? Motivation Suppose you needed to show a good bound for the gap between a fixed large constant, say$G=10^{10^{100}...
This is a speculation and perhaps naive. The theorem of Siegel that There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is ...