# Tagged Questions

For questions about approximating real numbers by rational numbers.

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### Numerators of close rational approximations to $\pi$

Define a rational number $\frac{p}{q}$ to be $\epsilon$-close to $\pi$ if $$\left|\pi - \frac{p}{q}\right| \leq \frac{1}{q^{\mu-\epsilon}}$$ where $\mu$ is the irrationality measure of $\pi$, ...
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### Siegel's Lemma for two solutions

Consider the homogeneous diophantine equation $$ax_1+bx_2+cx_3=0$$ over $\mathbb{Z}$ with a, b, c coprime. (A version of) Siegel's Lemma states, that there exists a non-trivial solution $x$, such ...
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### Show that $\inf \{ m+n\omega: m+n\omega>0,and,m,n\in{Z}\}= 0$, where $\omega>0$ is irrational.

Let $\omega\in\mathbb {R}$ be an irrational positive number. Set $$A=\{m+n\omega: m+n\omega>0,and,m,n\in{Z}\}.$$ Show that $\inf{A}=0.$ How should I start this problem? I don't get this problem.
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### Find the general solutions of $x^2+qy^2-z^2=4n$

I have this diophantine equation and i need the general solution form for it the equation $$x^2+qy^2-z^2=4n$$ some conditions $y\le 0,x\ge 0,z\ge 0$ x,z are even or odd together $n=5y+x-3xy$
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### positive integer solutions of $6xy+5x+5y=n$

Can any one help me with this? Determine all Positive integer $(x,y)$ such that $2 \le x \le y$ and $6xy+5x+5y=n$ Please do not solve as $(6x+5)(6y+5)=6n+5^2$, I need a more helpful method. we have ...
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### Non-constructive proofs for the rationality of a number

One of the key ideas in transcedental number theory is proving that a number is transcedental (i.e. not the root of any polynomial with integer coefficients) by showing a sequence of rational numbers ...
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### Hardy- Littlewood Circle Method

I'm currently trying to get to grips with the Hardy Littlewood circle method so I'm working through Vaughan's book. In the past I've been very bad for leaving a point behind if I don't follow it so I'...
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### Product of denominators exceed $n$ in Farey sequence

Why in the $n$-th Farey sequence the product of the denominators of $2$ adjacent fractions exceed $n$ ($0$ and $1$ are excluded) ? I have a theorem of Hurwitz which states: For every irrational ...
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### Infinitely many rationals with $|a-\frac pq|<\frac1{q^2}$

If $a$ is irrational, there are infinitely many $\frac pq$ s.t $|a-\frac pq|<\frac1{q^2}\tag1$ I have the proof but don't understand it: Take a finite set of rationals $S$ then for sufficiently ...
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### Rational approximation of square roots

I'm trying to find the best way to solve for rational approximations of the square root of a number, given some pretty serious constraints on the operations I can use to calculate it. My criteria for ...
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### Recursive definition of Minkowski ?(x) function

There is a fact that $?(\frac{a+b}{c+d}) = (?(\frac {a}{c}) + ?(\frac{b}{d})) / 2$ if a/c and b/d are adjanced elements of Farey sequence. How to prove it? I don't have any ideas at all.
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### Intuition in understanding Minkowski question mark ?(x) function

There are 3 definitions of Minkowski function: 1) If $[a_0; a_1, ...]$ is a continuous function representation of n 2) Consider the different ways of interpreting an infinite string of bits ...
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### Thue-Siegel-Roth Type Theorem

Dirichlet's approximation theorem says that for every real $\alpha$ and every positive integer $N$, there exist integers $p,q$ with $1 \leq q \leq N$ such that $$|q\alpha - p| < \frac{1}{N}.$$ ...
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### A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
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### Finding irrational numbers in given interval

If $~\xi~$ is irrational number then it is known that the set $~\{ p \xi + q ~ | ~ p,q \in \mathbb{Z} \}~$ is dense in $~\mathbb{R}$. Thus given some reals $~a~$ and $~b~$ one can find integers $~p~$ ...