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4
votes
1answer
722 views

Strongest known version of Baker's theorem

The article I have checked for Baker's theorem is Waldschmidt's. But the article and the citations therein are from the time of '88. Question: What is the the strongest known lower bound for ...
19
votes
3answers
2k views

When is $n/\ln(n)$ close to an integer?

As usual I expect to be critisised for "duplicating" this question. But I do not! As Gjergji immediately notified, that question was from numerology. The one I ask you here (after putting it in my ...
6
votes
1answer
314 views

Numbers characterized by extremal properties

The golden ratio $\phi=\frac{1+\sqrt5}2$ is sometimes said to be one of the most difficult numbers to approximate with rational numbers, because its continued fraction development $$\phi = 1 + ...
19
votes
2answers
1k views

How often are irrational numbers well-approximated by rationals?

Suppose $x\in \mathbb{R}$ is irrational, with irrationality measure $\mu=\mu(x)$; this means that the inequality $|x-\frac{p}{q}|< q^{-\lambda}$ has infinitely many solutions in integers $p,q$ if ...
67
votes
4answers
4k views

If 2^x and 3^x are integers, must x be as well?

I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number. If $n^x$ is an integer for ...