The diophantine-approximation tag has no wiki summary.

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### Is there a limit of cos (n!)?

Hi,
I encountered a problem today to prove that cos (n!) does not have a limit. I have no idea how to do it formally. Could someone help? The simpler the proof (by that i mean less complex theorems ...

**8**

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**1**answer

367 views

### is there any way to bound the number of CM points by height functions?

It is known that if $X$ is a curve over a number field $F$ equipped with a flat regular model over $O_F$ the integer ring, one can define, using a suitable ample line bundle with an Hermitian metric, ...

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### What numbers can be approximated “pretty well” by rationals?

More precisely, what real numbers $r$ have the following property: for any $\epsilon > 0$ there exist infinitely many pairs $(p, q)$ of integers such that
$$\left| \frac{p}{q} - r \right| < ...

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**1**answer

346 views

### An asymptotic expression for the solution to the squares problem suggested by statistical mechanics

The $s$ squares problem is to count the number $r_s (n)$ of integer solutions $(x_1,x_2,...,x_s)$ of the Diophantine equation $x_{1}^{2}+x_{2}^{2}+...+x_{s}^{2}=n$ in which changing the sign or ...

**6**

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### Can you get Siegel's theorem “for free” from modularity and Mazur's Eisenstein Ideal paper?

There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...

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**2**answers

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### Simultaneous rational approximation of two reals using their continued fractions

Littlewood's well-known conjecture about simultaneous rational approximation is that for all $x, y \in \mathbb{R}$, $\liminf_{n \to \infty} n \Vert nx \Vert \Vert ny \Vert = 0$ (where $\Vert x \Vert$ ...

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**3**answers

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### Striking applications of Baker's theorem

I saw that there are many "applications" questions in Mathoverflow; so hopefully this is an appropriate question. I was rather surprised that there were only five questions at Mathoverflow so far with ...

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**1**answer

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### 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**

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**3**answers

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**

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**1**answer

318 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 + ...

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**1**answer

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### 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 ...

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**4**answers

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### 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 ...