Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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Solution of diophantine equation with lowest c

Lets say I have a diophantine equation , aX - bY = c Now, for some (a,b,c) I may not have any integer solution at all. But lets say , I write the equation in this way , aX - bY = c + p p is an integer . (positive or negative) So, I can increase the…
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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.
sooobus
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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 $$ |q\alpha - p| < \frac{1}{N}. $$ It follows that for every real $\alpha$, there are…
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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 number $\alpha\in\mathbb R$ there exists infinitely…
user1161
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Representations of some primes as $x^2-2y^2$?

I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings: $$ p\equiv\pm1(\mod8)\longrightarrow p=x^2-2y^2 $$ Any help appreciated.
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How to find cases where $m^2$ is near to $2^A$?

In another problem here in MSE I ran into the question how I can (practically, in a program) find (positive) integer $m$ such that they are "near" to perfect powers of $2$, so $$ (0 \lt ) \qquad d_m = 2^{A_m} - m^2 \qquad \qquad \text{ is "small"}…
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measure of $|\alpha-\frac pq|\lt\frac1{4q^2}$ with infinitely solutions

$\alpha\in[0,1]$, and $$|\alpha-\frac pq|\lt\frac1{4q^2}$$ has infinitely solutions $p, q\in\Bbb Z$, $\gcd(p,q)=1$. Let $E$ be the set of all such $\alpha\in[0,1]$, that is $$E=\{\alpha\in[0,1]\colon |\alpha-\frac pq|\lt\frac1{4q^2} \text{ has…
2016
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using kronecker's theorem can we prove there's some power of two yielding a number whose initial digits equal my social security number?

I just watched the "Great Courses" series of lectures in number theory, in which Professor Burger stated that using Kronecker's theorem for any irrational number r, the sequence ({n * r}) where n >= 0 is dense in the interval [0,1) we can…
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On Diophantine approximation and irrationality proofs

This question is an offshoot from this previous MSE post. I have a ratio of two numbers $a$ and $b$ (presumably both positive integers), where $a$ and $b$ are determined by some arithmetic / number-theoretic equation $f(a,b)=0$. Let $S$ be the…
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On a theorem of Kronecker!

Let $\alpha$ be an irrational number and $\beta$ be an arbitrary real number, Prove that there are infinitely many pair of integers $(x,y)$ with $x\in\mathbb{N}$ such that: $$|x\alpha-y-\beta|<\frac3x$$ This theorem is due to polish mathematician…
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How to find the fraction of integers with the smallest denominator matching an interval?

I have a real interval $\mathbb{S} \subset ]0; 1[$. How can I find the smallest $n \in \mathbb{N}$, for which there is a $k \in \mathbb{N}$, for which $\frac{k}{n} \in \mathbb{S}$? 1The important thing is to do this effectively. For example, just…
peterh
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A series of positive terms to prove $\pi>\frac{333}{106}$

This is a consequence of the answer to that question. A proof that $\pi > \frac{333}{106}$ is given by the series of positive terms $$\pi-\frac{333}{106} \\ =\frac{48}{371} \sum_{k=0}^\infty \frac{118720 k^2+762311 k+1409424}{(4 k+9) (4 k+11) (4…
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