# 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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### 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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### 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 $~p~$ ...
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### Near-integer solutions to $y=x(1+2\sqrt{2})$ given by sequence - why?

EDIT: I've asked the same basic question in its more progressed state. If that one gets answered, I'll probably accept the answer given below (although I'm uncertain of whether or not this is the ...
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### Question on a constructive proof of irrationality of $\sqrt 2$

Here is the constructive proof of $\sqrt 2 \not \in \mathbb Q$ found on this page : Given positive integers $a$ and $b$, because the valuation (i.e., highest power of 2 dividing a number) of $2b^2$...
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### Average of a certain diophantine function

Yesterday I was browsing math.se and came across this question. It was answered by a few people and the best answer was already accepted so I just read the question and the solutions to it. Then I ...
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### Three-gap problem, easy version.

Let $N$ be a positive integer and $\theta$ an angle in $(0, 2\pi)$. Consider the map$$f: \{0, 1, 2, \dots, N-1, N\} \to \text{unit circle}, \text{ }f(k) = k\theta \text{ }(\text{mod } 2\pi).$$Show ...
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### Specification of Hurwitz's Theorem

Hurwitz's Theorem in Number Theory states that for every irrational number $\xi$, the equation $$\left|\xi-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}$$ has infinitely many solutions $(p,q)$ ...
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### Show that $\left| \sqrt2-\frac{h}{k} \right| \geq \frac{1}{4k^2},$ for any $k \in \mathbb{N}$ and $h \in \mathbb{Z}$.
Show that $$\left| \sqrt2-\frac{h}{k} \right| \geq \frac{1}{4k^2},$$ for any $k \in \mathbb{N}$ and $h \in \mathbb{Z}$. I tried many different ways to expand left side and estimate it but always got ...
### Prove that $p_nq_{n-1}$ - $p_{n-1}q_n=(-1)^{n-1}$ for $p_{-2}=0$ $p_{-1}=1$ $q_{-2}=1$ $q_{-1}=0$
Let $p_n$/$q_n$ for $n=0,1,2,..$ be the convergents of $a∈ R$ $p_{-2}=0$ $p_{-1}=1$ $q_{-2}=1$ $q_{-1}=0$ $p_n= a_np_{n-1}+p_{n-2}$ $q_n= a_nq_{n-1}+q_{n-2}$ I need to prove that \$p_nq_{n-1} - p_{...