Number theory/Signed Articles/Elementary diophantine approximations

The theory of diophantine approximations is a chapter of number theory, which in turn is a part of mathematics. It studies the approximations of real numbers by rational numbers. This article presents an elementary introduction to diophantine approximations, as well as an introduction to number theory via diophantine approximations.

Introduction

In the everyday life our civilization applies mostly (finite) decimal fractions $\ {\frac {k}{10^{n}}}.$  Decimal fractions are used both as certain values, e.g. $5.85, and as approximations of the real numbers, e.g. $\ \pi \approx 3.14159.$  However, the field of all rational numbers is much richer than the ring of the decimal fractions (or of the binary fractions $\ {\frac {k}{2^{n}}},$  which are used in the computer science). For instance, the famous approximation $\ \pi \approx {\frac {355}{113}}$  has denominator 113 much smaller than 10⁵ but it provides a better approximation than the decimal one, which has five digits after the decimal point.

How well can real numbers (all of them or the special ones) be approximated by rational numbers? A typical Diophantine approximation result states:

Theorem  Let $\ a$  be an arbitrary real number. Then

• $\ a$  is rational   $\Leftrightarrow$   there exists a real number C > 0 such that

$\left|a-{\frac {x}{y}}\right|>{\frac {C}{y}}$

for arbitrary integers $\ (x,y)$  such that $\ y>0$  and $\ a\neq {\frac {x}{y}};$

• (Adolph Hurwitz)   $\ a$  is irrational   $\Leftrightarrow$   there exist infinitely many pairs of integers $\ (x,y)$  such that $\ y>0$  and

$\left|a-{\frac {x}{y}}\right|<{\frac {1}{{\sqrt {5}}\cdot y^{2}}}.$

Remark  Implication   $\Leftarrow$   of the first part of the theorem is a simple and satisfaction bringing exercise.

Notation

• $\Leftarrow :\Rightarrow \$   —   "equivalent by definition" (i.e. "if and only if");
• $:=\$   —   "equals by definition";
• $\exists$   —   "there exists";
• $\forall$   —   "for all";
• $a\in A\$   —   "$a\$  is an element of set $\ A$";

 

• $\mathbb {N} \ :=\ \{1,2\dots \}$  —  the semiring of the natural numbers;
• $\mathbb {Z} _{+}\ :=\ \{0,1,\dots \}$  —  the semiring of the non-negative integers;
• $\mathbb {Z} \ :=\ \{-2,-1,1,2\dots \}$  —  the ring of integers;
• $\mathbb {Q}$  —  the field of rational numbers;
• $\mathbb {R}$  —  the field of real numbers;

 

• $(x;y)\ :=\{t\in \mathbb {R} :\,x<t<y\}$
• $|(x;y)|\ :=y-x$
• ${\mathit {Span}}(x,y)\ :=\ \left(\min \left(x,y\right);\,\max \left(x,y\right)\right)$

 

• $x|y\$   —   "$x\$  divides $\ y$"   (i.e. ${\frac {y}{x}}\in \mathbb {Z}$);
• $\gcd(a,b)\$  —  the greatest common divisor of integers $\ a$  and $\ b.$

 

The method of neighbors and median

In this section we will quickly obtain some results about approximating irrational numbers by rational. To this end we will not worry about the details of the difference between a rational number and a fraction (with integer numerator and denominator)—this will not cause any problems; fully crisp notions will be developed in the next sections, they will involve 2-dimensional vectors and 2x2 matrices. This section is still introductory. It is supposed to provide quick insight into the topic.

Definitions

Fractions ${\frac {a}{c}}$  and ${\frac {b}{d}},$  with integer numerators and natural denominators, are called neighbors (in the given order)   $\Leftarrow :\Rightarrow$

${\frac {a}{c}}-{\frac {b}{d}}\ =\ {\frac {1}{c\cdot d}}$

Fraction ${\frac {a}{c}}$  is called the top neighbor of the other, ${\frac {b}{d}}$  is called the bottom neighbor, and the interval $\ {\mathit {Span}}({\frac {a}{c}},{\frac {b}{d}})$  is called neighborhood; thus a neighborhood is an open interval such that its endpoints are neighbors.

• If  ${\frac {a}{c}}$  and ${\frac {b}{d}}$  are neighbors then  $\ a\cdot b\geq 0$   ( i.e.  ${\frac {a}{c}}\cdot {\frac {b}{d}}\ \geq 0$ ).

• Let $\ a,b,c,d\in \mathbb {N} .$   Fractions ${\frac {a}{c}}$  and ${\frac {b}{d}}$  are neighbors   $\Leftrightarrow$   fractions ${\frac {d}{b}}$  and ${\frac {c}{a}}$  are neighbors   $\Leftrightarrow$   fractions ${\frac {-b}{d}}$  and ${\frac {-a}{c}}$  are neighbors.

Examples:

• Fractions ${\frac {n+1}{1}}$  and ${\frac {n}{1}}$  are neighbors for every positive integer $\ n.$

• Fractions ${\frac {1}{n}}$  and ${\frac {1}{n+1}}$  are neighbors for every positive integer $\ n.$

Thus it easily follows that for every positive irrational number $\ x$  there exists a pair of neighbors  ${\frac {a}{c}}$  and  ${\frac {b}{d}},$  with positive numerators and denominators, such that:

${\frac {a}{c}}\ >\ x\ >\ {\frac {b}{d}}$ 

Definition  A pair of neighboring fractions $\left({\frac {a}{c}},{\frac {b}{d}}\right),$  with integer numerators and natural denominators, is called a top pair   $\Leftarrow :\Rightarrow \ c>d.$   Otherwise it is called a bottom pair.

Thus now we have notions of top/bottom neighbors and of top/bottom pairs of neighbors.

• Let  $\left({\frac {a}{c}},{\frac {b}{d}}\right),$  be a pair of neighbors. Then  $\left({\frac {a}{c}},{\frac {a+b}{c+d}}\right)$  is a top pair of neighbors, and  $\left({\frac {a+b}{c+d}},{\frac {b}{d}}\right)$  is a bottom pair of neighbors.

 

First results

Theorem  Let fractions ${\frac {a}{c}}$  and ${\frac {b}{d}},$  with integer numerators and natural denominators, be neighbors. Then

• if integers $\ t>0$  and $\ s$  are such that   $\ {\frac {a}{c}}>{\frac {s}{t}}>{\frac {b}{d}},$   then   $t\geq c+d;$

• the median  ${\frac {a+b}{c+d}}$  is a bottom neighbor of  ${\frac {a}{c}},$  and a top neighbor of  ${\frac {b}{d}};$ 

• let  $\ x$  be an irrational number such that   ${\frac {a}{c}}\ >\ x\ >\ {\frac {b}{d}};$   then

$0\ <\ \max({\frac {a}{c}}-x,\,x-{\frac {b}{d}})\ <\ {\frac {1}{c\cdot d}}$

and

$0\ <\ {\frac {a}{c}}-x\ <\ {\frac {1}{2\cdot c^{2}}}$       or       $0\ <\ x-{\frac {b}{d}}\ <\ {\frac {1}{2\cdot d^{2}}}$

Proof   Let   $\ {\frac {a}{c}}>{\frac {s}{t}}>{\frac {b}{d}};$   then

${\frac {a}{c}}-{\frac {s}{t}}\ \geq \ {\frac {1}{c\cdot t}}$     and     ${\frac {s}{t}}-{\frac {b}{d}}\ \geq \ {\frac {1}{d\cdot t}}$

and

${\frac {1}{c\cdot t}}+{\frac {1}{d\cdot t}}\ \leq {\frac {1}{c\cdot d}}$

Multiplying this inequality by  $c\cdot d\cdot t\$ gives

$t\geq c+d$

which is the first part of our theorem.

The second part of the theorem is obtained by a simple calculation, straight from the definition of the neighbors.

The first inequality of the third part of the theorem is instant:

$\max({\frac {a}{c}}-x,\,x-{\frac {b}{d}})\ <\ ({\frac {a}{c}}-x)+(x-{\frac {b}{d}})\ =\ {\frac {1}{c\cdot d}}$

Next,

$\left({\frac {1}{c}}-{\frac {1}{d}}\right)^{2}\ \geq \ 0$

hence

${\frac {1}{2\cdot c^{2}}}+{\frac {1}{2\cdot d^{2}}}\ \geq \ {\frac {1}{c\cdot d}}\ =\ {\frac {a}{c}}-{\frac {b}{d}}$

$=\ \left|{\mathit {Span}}({\frac {a}{c}},{\frac {b}{d}})\right|$

and

${\frac {b}{d}}+{\frac {1}{2\cdot d^{2}}}\ \geq \ {\frac {a}{c}}-{\frac {1}{2\cdot c^{2}}}$

i.e.

${\mathit {Span}}\left({\frac {a}{c}},{\frac {a}{c}}-{\frac {1}{2\cdot c^{2}}}\right)\ \cup \ {\mathit {Span}}\left({\frac {b}{d}}+{\frac {1}{2\cdot d^{2}}},{\frac {b}{d}}\right)\ \backslash \ \mathbb {Q} \ \supseteq \ {\mathit {Span}}\left({\frac {a}{c}},{\frac {b}{d}}\right)\ \backslash \ \mathbb {Q}$

End of proof

Corollary  Let fractions ${\frac {a}{c}}$  and ${\frac {b}{d}},$  with integer numerators and natural denominators, be neighbors. Then, if integers $\ t>0$  and $\ s$  are such that   $\ {\frac {a}{c}}>{\frac {s}{t}}>{\frac {b}{d}},$   then either

• $s=a+b\quad \land \quad t=c+d;$

or

• $t\ \geq \ c+d+\min(c,d)\ >\ c+d$

 

Hurwitz theorem

Let  $\ x\in \mathbb {R}$  be an arbitrary irrational number. Then

$\left|x-{\frac {s}{t}}\right|\ <\ {\frac {1}{{\sqrt {5}}\cdot t^{2}}}$

for infinitely many different $\ s,t\in \mathbb {Z} \backslash \{0\}.$

Lemma 1

Let  $\ c,d\in \mathbb {N} .$  Let  $\ C:=c+d.$  Then:

$c^{2}-{\sqrt {5}}\!\cdot \!c\!\cdot \!d+d\,^{2}\ >\ 0$

or

$C\,^{2}-{\sqrt {5}}\cdot \!C\!\cdot \!d+d\,^{2}\ >\ 0$

Proof of lemma 1  It's easy to show that  ${\sqrt {2}}\cdot c\neq {\sqrt {3-{\sqrt {5}}}}\cdot d.$  Thus the square of  $\ X:={\sqrt {2}}\cdot c-{\sqrt {3-{\sqrt {5}}}}\cdot d$  is positive. Now,

${\sqrt {2}}\cdot {\sqrt {3-{\sqrt {5}}}}\ =\ {\sqrt {5}}-1$

which means that we may write  $\ X^{2}$  as follows:

$2\!\cdot \!c^{2}\ -\ 2\!\cdot \!({\sqrt {5}}-1)\cdot c\!\cdot \!d\ +\ (3-{\sqrt {5}})\!\cdot \!d\,^{2}\ >\ 0$

i.e.

$(c^{2}-{\sqrt {5}}\cdot c\cdot d+d^{2})\ +\ (C^{2}-{\sqrt {5}}\cdot C\cdot d+d^{2})\ >\ 0$

and lemma 1 follows.  End of proof

Lemma 2

Let  $\ a,b\in \mathbb {Z}$  and  $\ c,d\in \mathbb {N} .$  Let  $\ A:=a+b$  and  $\ C:=c+d.$  Furthermore, let fractions  ${\frac {a}{c}},{\frac {b}{d}},$  be neighbors, and let:

${\frac {A}{C}}\ >\ x\ >\ {\frac {b}{d}}$

where $\ x$  is real. Then one of the following three inequalities holds:

• $0\ <\ {\frac {a}{c}}-x\ <\ {\frac {1}{{\sqrt {5}}\cdot c^{2}}}$

• $0\ <\ {\frac {A}{C}}-x\ <\ {\frac {1}{{\sqrt {5}}\cdot C^{2}}}$

• $0\ <\ x-{\frac {b}{d}}\ <\ {\frac {1}{{\sqrt {5}}\cdot d^{2}}}$

Proof  There are two cases along the inequalities of Lemma 1. Let's assume the first one, which is equivalent to:

${\frac {1}{{\sqrt {5}}\cdot c^{2}}}+{\frac {1}{{\sqrt {5}}\cdot d\,^{2}}}\ >\ {\frac {1}{c\cdot d}}\ =\ {\frac {a}{c}}-{\frac {b}{d}}\ =\ \left|{\mathit {Span}}\left({\frac {a}{c}},{\frac {b}{d}}\right)\right|$

Thus

${\frac {b}{d}}+{\frac {1}{{\sqrt {5}}\cdot d\,^{2}}}\ >\ {\frac {a}{c}}-{\frac {1}{{\sqrt {5}}\cdot c^{2}}}$

which means that

${\mathit {Span}}\left({\frac {a}{c}},{\frac {a}{c}}-{\frac {1}{{\sqrt {5}}\cdot c^{2}}}\right)\ \cup \ {\mathit {Span}}\left({\frac {b}{d}}+{\frac {1}{{\sqrt {5}}\cdot d\,^{2}}},{\frac {b}{d}}\right)\ \supseteq \ {\mathit {Span}}\left({\frac {a}{c}},{\frac {b}{d}}\right)$

$\supseteq \ {\mathit {Span}}\left({\frac {A}{C}},{\frac {b}{d}}\right)$

Thus lemma 2 is proven when the first inequality of lemma 1 holds. By replacing in the above proof the lower case  $\ a,c,$  by the upper case  $\ A,C,$  we obtain the proof when the second inequality of lemma 1 holds.

End of proof (of lemma 2)

Lemma 2'

Let  $\ a,b\in \mathbb {Z}$  and  $\ c,d\in \mathbb {N} .$  Let  $\ A:=a+b$  and  $\ C:=c+d.$  Furthermore, let fractions  ${\frac {a}{c}},{\frac {b}{d}},$  be neighbors, and let:

${\frac {a}{c}}\ >\ x\ >\ {\frac {A}{C}}$

where $\ x$  is real. Then one of the following three inequalities holds:

• $0\ <\ x-{\frac {b}{d}}\ <\ {\frac {1}{{\sqrt {5}}\cdot d\,^{2}}}$

• $0\ <\ x-{\frac {A}{C}}<\ {\frac {1}{{\sqrt {5}}\cdot C^{2}}}$

• $0\ <\ {\frac {a}{c}}\ <\ {\frac {1}{{\sqrt {5}}\cdot x^{2}}}$

Proof  It's similar to the proof of lemma 2. Or one may apply lemma 2 to  $\ x':=-x,\ a':=-b,\ b':=-a,$  and  $\ c':=d,\ d':=c$, which would provide us with the respective fraction  ${\frac {s'}{t'}}.$  Then the required  $\ s,t,$  are given by  $s:=-s',\ t:=t'.$

End of proof (of lemma 2')

 

Proof of Hurwitz theorem

When  $\ x$  is irrational, then it is squeezed between infinitely many different pairs of neighbors (see part two of the theorem from the First results section). They provide infinitely many different required fractions  ${\frac {s}{t}}$  (see lemma 2 and 2' above; but it is not claimed, nor true, that different pairs of neighbors provide different required fractions).

End of proof

 

Squeezing irrational numbers between neighbors

Let $\ x>0$  be an irrational number, We may always squeeze it between the extremal neighbours:

${\frac {1}{0}}\ >\ x\ >\ {\frac {0}{1}}$

But if you don't like infinity (on the left above) then you may do one of the two things:

$x>1\quad \quad \Rightarrow \quad \quad {\frac {n+1}{1}}>x>{\frac {n}{1}}$

or

$x<1\quad \quad \Rightarrow \quad \quad {\frac {1}{n}}>x>{\frac {1}{n+1}}$

where in each of these two cases  $\ n\in \mathbb {N}$   is a respective unique positive integer.

It was mentioned in the previous section (First results) that if fractions ${\frac {a}{c}}$  and ${\frac {b}{d}},$  with positive (or non-negative) integer numerators and denominators are neighbors then also the top and the bottom (bot  for short) pair:

• ${\mathit {top}}\!\left({\frac {a}{c}},{\frac {b}{d}}\right)\ :=\ \left({\frac {a}{c}},{\frac {a+b}{b+d}}\right)$

and

• ${\mathit {bot}}\!\left({\frac {a}{c}},{\frac {b}{d}}\right)\ :=\ \left({\frac {a+b}{c+d}},{\frac {b}{d}}\right)$

are both pairs of neighbors.

Let $\ A_{0}$  be a pair of neighbors, and $\ x\in {\mathit {Span}}(A_{0})$ be an irrational number. Assume that pairs of neighbors $\ A_{0},\dots ,A_{n-1}$  are already defined, and that they squeeze $\ x,$ i.e. that $\ x\in {\mathit {Span}}(A_{k})$  for each $\ k=0,\dots ,n-1.$  Then we define $\ A_{n}$  as the one of the two pairs: $\ {\mathit {top}}(A_{n-1})$  or  $\ {\mathit {bot}}(A_{n-1}),$  which squeezes $\ x.$  Thus for every positive irrational number we have obtained an infinite sequence of pairs of neighbors, each squeezing the given irrational number more and more. Thus for arbitrary irrational $\ x$  there exist fractions of integers $\ {\frac {s}{t}},$  with arbitrarily large denominators, such that

$\left|x-{\frac {s}{t}}\right|\ <\ {\frac {1}{{\sqrt {5}}\cdot t^{2}}}$

(see section Hurwitz theorem).

If cases top-top and bot-bot happen only finitely many times then starting with an $\ n$  we get an infinite alternating top-bot-top-bot-... sequence:

$A_{n+1}={\mathit {top}}(A_{n}),\quad A_{n+2}={\mathit {bot}}(A_{n+2}),\quad A_{n+3}={\mathit {top}}(A_{n+1})\quad \dots$

Then the new neighbor of the $\ A_{n+k}$  pair (i.e. the median of the previous pair $\ A_{n+k-1}$)  is equal to

$e_{k}\ :=\ {\frac {F_{k+1}\cdot a+F_{k}\cdot b}{F_{k+1}\cdot c+F_{k}\cdot d}}$

for every $\ k=1,2,\dots ,$  where $\ F_{t}$  are the Fibonacci numbers, where

$\left({\frac {a}{c}},{\frac {b}{d}}\right)\ :=\ A_{n}$

It is known that

$\lim _{k\rightarrow \infty }{\frac {F_{k+1}}{F_{k}}}\ =\ \Phi \ :=\ {\frac {{\sqrt {5}}+1}{2}}$

hence

$x\ =\ \lim _{k\rightarrow \infty }e_{k}\ =\ {\frac {\Phi \cdot a+b}{\Phi \cdot c+d}}$

But if our infinite alternation has started with bot :

$A_{n+1}={\mathit {bot}}(A_{n}),\quad A_{n+2}={\mathit {top}}(A_{n+2}),\quad A_{n+3}={\mathit {bot}}(A_{n+1})\quad \dots$

Then we would have

$x\ =\ \lim _{k\rightarrow \infty }e_{k}\ =\ {\frac {a+\Phi \cdot b}{c+\Phi \cdot d}}$

 

Another proof of Hurwitz Theorem (further insight)

Reduction to x > 0

Since

$\left|(-x)-{\frac {-s}{t}}\right|\ =\ \left|x-{\frac {s}{t}}\right|$

it is enough to prove Hurwitz theorem for positive irrational numbers only.

Two cases

Consider the sequence $\ \left(A_{n}\right)$  of pairs of neighbors, which squeeze $\ x>0,$  from the previous section. The case of the infinite alternation top-bot-top-bot-... has been proved already. In the remaining case the bot-bot-top or top-top-bot progressions appear infinitely many times, i.e. there are infinitely many non-negative integers $\ n$  for which

${\frac {a}{c}}\ >\ {\frac {a+b}{c+d}}\ >\ {\frac {a+2\cdot b}{c+2\cdot d}}\ >\ x\ >\ {\frac {a+3\cdot b}{c+3\cdot d}}\ >\ {\frac {b}{d}}$

or

${\frac {a}{c}}\ >\ {\frac {3\cdot a+b}{3\cdot c+d}}\ >\ x\ >\ {\frac {2\cdot a+b}{2\cdot c+d}}\ >\ {\frac {a+b}{c+d}}\ >\ {\frac {b}{d}}$

holds, where

$\left({\frac {a}{c}},{\frac {b}{d}}\right)\ :=\ A_{n}$

 

The top-top-bot case

Let's consider the latter top-top-bot case. Let  $\ \xi :={\frac {d}{c}}.$  The squeeze by neighbors:

${\frac {a}{c}}\ >\ x\ >\ {\frac {2\cdot a+b}{2\cdot c+d}}$

shows that

$0\ <\ {\frac {a}{c}}-x\ <\ {\frac {1}{c\cdot (2\cdot c+d)}}\ =\ {\frac {1}{(2+\xi )}}\cdot {\frac {1}{c^{2}}}$

This inequality provides the first insight (otherwise, we are not going to use it), so it deserves to be written cleanly as an implication:

$0\ <\ {\frac {a}{c}}-x\ <\ {\frac {1}{(2+\xi )}}\cdot {\frac {1}{c^{2}}}\quad \quad \Leftarrow \quad \quad {\frac {a}{c}}\ >\ x\ >\ {\frac {2\cdot a+b}{2\cdot c+d}}$

The relevant neighborhoods

Consider the next two pairs of neighbors, pair $\ A_{n+4}$  and pair $\ A_{n+5},$  which squeeze $\ x.$  The relevant neighborhoods are:

• $A\ :=\ {\mathit {Span}}({\frac {a}{c}},\,{\frac {3\cdot a+b}{3\cdot c+d}})$

• $B\ :=\ {\mathit {Span}}({\frac {3\cdot a+b}{3\cdot c+d}},\,{\frac {5\cdot a+2\cdot b}{5\cdot c+2\cdot d}})$

• $C\ :=\ {\mathit {Span}}({\frac {5\cdot a+2\cdot b}{5\cdot c+2\cdot d}},\,{\frac {7\cdot a+3\cdot b}{7\cdot c+3\cdot d}})$

• $D\ :=\ {\mathit {Span}}({\frac {7\cdot a+3\cdot b}{7\cdot c+3\cdot d}},\,{\frac {2\cdot a+b}{2\cdot c+d}})$

 

Neighborhood B

Let  $\ x\in B.$  Then

${\frac {a}{c}}\ >\ {\frac {3\cdot a+b}{3\cdot c+d}}\ >\ x\ >\ {\frac {5\cdot a+2\cdot b}{5\cdot c+2\cdot d}}\ >\ {\frac {2\cdot a+b}{2\cdot c+d}}$

and

$0\ <\ {\frac {a}{c}}-x\ <\ {\frac {1}{c\cdot (3\cdot c+d)}}+{\frac {1}{(3\cdot c+d)\cdot (5\cdot c+2\cdot d)}}$

$=\ {\frac {2}{c\cdot (5\cdot c+2\cdot d)}}\ =\ {\frac {2}{(5+2\cdot \xi )}}\cdot {\frac {1}{c^{2}}}$

Conclusion:

$0\ <\ {\frac {a}{c}}-x\ <\ {\frac {2}{5+2\cdot \xi }}\cdot {\frac {1}{c^{2}}}\quad \quad \Leftarrow \quad \quad x\in B$

Neighborhood C (first C-inequality)

Let  $\ x\in C.$  Then

${\frac {a}{c}}\ >\ {\frac {3\cdot a+b}{3\cdot c+d}}\ >\ {\frac {5\cdot a+2\cdot b}{5\cdot c+2\cdot d}}\ >\ x\ >\ {\frac {7\cdot a+3\cdot b}{7\cdot c+3\cdot d}}$

Thus using the calculation for neighborhood B also for C, we get

$0\ <\ {\frac {a}{c}}-x\ <\ {\frac {2}{(5+2\cdot \xi )\cdot c^{2}}}+{\frac {1}{(5\cdot c+2\cdot d)\cdot (7\cdot c+3\cdot d)}}$

$=\ {\frac {3}{7+3\cdot \xi }}\cdot {\frac {1}{c^{2}}}$

First C-conclusion:

$0\ <\ {\frac {a}{c}}-x\ <\ {\frac {3}{7+3\cdot \xi }}\cdot {\frac {1}{c^{2}}}\quad \quad \Leftarrow \quad \quad x\in C$

 

Neighborhood D

Let  $\ x\in D,$  i.e.

${\frac {7\cdot a+3\cdot b}{7\cdot c+3\cdot d}}\ >\ x\ >\ {\frac {2\cdot a+b}{2\cdot c+d}}$

Let  $\alpha :=2\cdot a+b$   and   $\gamma :=2\cdot c+d=(2+\xi )\cdot c.$  Then

$0\ <\ x-{\frac {\alpha }{\gamma }}\ =\ x-{\frac {2\cdot a+b}{2\cdot c+d}}$

$<\ {\frac {1}{(7\cdot c+3\cdot d)\cdot (2\cdot c+d)}}\ =\ {\frac {1}{(7+3\cdot \xi )\cdot c\cdot \gamma }}$

$=\ {\frac {2+\xi }{7+3\cdot \xi }}\cdot {\frac {1}{\gamma ^{2}}}$

Conclusion:

$0\ <\ x-{\frac {\alpha }{\gamma }}\ <\ {\frac {2+\xi }{7+3\cdot \xi }}\cdot {\frac {1}{\gamma ^{2}}}\quad \quad \Leftarrow \quad \quad x\in D$

Early yield (Hurwitz Theorem)

Let's impatiently indulge ourselves in already getting some crude results from the above hard work (:-). The above three BCD-conclusions instantly imply:

• $0\ <\ {\frac {a}{c}}-x\ <\ {\frac {2}{5}}\cdot {\frac {1}{c^{2}}}\quad \quad \Leftarrow \quad \quad x\in B$

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\ <\ \frac{a}{c}-x\ <\ \frac{3}{7}\cdot\frac{1}{c^2}\quad\quad\Leftarrow\quad\quad x\in C}

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\ <\ x-\frac{\alpha}{\gamma}\ <\ \frac{2}{7}\cdot\frac{1}{\gamma^2}\quad\quad\Leftarrow\quad\quad x\in D}

Since

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \max(\frac{2}{5},\,\frac{3}{7},\,\frac{2}{7})\ =\ \frac{3}{7}\ <\ \frac{1}{\sqrt{5}},}

each occurrence of the top-top-bot subsequence, i.e. of equalities:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{n+1} = \mathit{top}(A_n),\quad A_{n+2} = \mathit{top}(A_{n+1}),\quad A_{n+3} = \mathit{bot}(A_{n+2})\quad\dots}

provides a fraction  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{s}{t}\in \mathit{Span}(A_{n+3}),}   with integer numerator and natural denominator, such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|x-\frac{s}{t}\right|\ <\ \frac{3}{7}\cdot\frac{1}{t^2}\ <\ \frac{1}{\sqrt{5}}\cdot\frac{1}{t^2}}

The same is holds for every occurrence of the bot-bot-top sequence, which can be shown now mechanically by a proof similar to the proof of the top-top-bot case, or as follows: define the squeezing sequence of  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ -x}   by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_n := (\frac{-b}{d},\frac{-a}{c})}     for     Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_n = (\frac{a}{c},\frac{b}{d})}

Let  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(A_n,A_{n+1},A_{n+2},A_{n+3}\right)}   be a bot-bot-top progression. Then  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(B_n,B_{n+1},B_{n+2},B_{n+3}\right)}   is a top-top-bot progression which squeezes  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ -x.} Thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|(-x)-\frac{u}{w}\right|\ <\ \frac{3}{7}\cdot\frac{1}{w^2}\ <\ \frac{1}{\sqrt{5}}\cdot\frac{1}{w^2}}

for certain  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{u}{w}\in \mathit{Span}(B_{n+3}),}   with integer numerator and natural denominator. Then  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{s}{t}\in\mathit{Span}(A_{n+3}),}   for  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (s,t):=(-u,w),}   satisfies:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|x-\frac{s}{t}\right|\ <\ \frac{3}{7}\cdot\frac{1}{t^2}\ <\ \frac{1}{\sqrt{5}}\cdot\frac{1}{t^2}}

When another top-top-bot or bot-bot-top progression starts with a sufficiently large index  $\ n',$  then  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\mathit{Span}(A_{n'+3})\right| < \left|x-\frac{s}{t}\right|,}   which means that the respective new approximation  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{s'}{t'}\in\mathit{Span}(A_{n'+3})}   is different. It follows that if there are infinitely many progressions top-top-bot or bot-bot-bot then there are infinitely many fractions  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \frac{s}{t},} which satisfy the inequality above. Thus we have obtained the following version of Hurwitz theorem:

Theorem  Let  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in\mathbb{R}\backslash\mathbb{Q}}   be an arbitrary irrational number. Then inequality

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|x-\frac{s}{t}\right|\ <\ \frac{1}{\sqrt{5}}\cdot\frac{1}{t^2}}

holds for infinitely many fractions  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \frac{s}{t},}   with integer numerator and natural denominator. Furthermore, if the squeezing sequence of  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x} does not eventually alternate top-bot-top-bot-... till infinity, i.e. if it has infinitely many top-top or bot-bot progressions, then

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|x-\frac{s}{t}\right|\ <\ \frac{3}{7 }\cdot\frac{1}{t^2}}

holds for infinitely many fractions  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \frac{s}{t},}   with integer numerator and natural denominator.

 

Neighborhood C (second C-inequality)

Let  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x\in C.}   Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{5\cdot a+2\cdot b}{5\cdot c+2\cdot d}\ >\ x\ >\ \frac{7\cdot a+3\cdot b}{7\cdot c+3\cdot d}\ >\ \frac{2\cdot a+b}{2\cdot c+d}}

Thus, using the earlier conclusion for neighborhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ D}   also for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ C,}   we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\ <\ x-\frac{\alpha}{\gamma}\ =\ x-\frac{2\cdot a+b}{2\cdot c+d}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <\ \frac{1}{(5\cdot c+2\cdot d)\cdot(7\cdot c+3\cdot d)}\ +\ \frac{2+\xi}{(7+3\cdot\xi)\cdot \gamma^2}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\ \frac{1}{(5+2\cdot\xi)\cdot(7+3\cdot\xi)}\cdot\frac{1}{c^2}\ +\ \frac{2+\xi}{(7+3\cdot\xi)\cdot \gamma^2}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\ \frac{(2+\xi)^2}{(5+2\cdot\xi)\cdot(7+3\cdot\xi)}\cdot\frac{1}{\gamma^2}\ +\ \frac{2+\xi}{(7+3\cdot\xi)\cdot \gamma^2}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\ \frac{2+\xi}{5+2\cdot\xi}\cdot\frac{1}{\gamma^2}}

Second C-conclusion:

$0\ <\ x-{\frac {\alpha }{\gamma }}\ <\ {\frac {2+\xi }{5+2\cdot \xi }}\cdot {\frac {1}{\gamma ^{2}}}\quad \quad \Leftarrow \quad \quad x\in C$

Neigborhood C (the combined inequality)

Let:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi_0\ :=\ \frac{\sqrt{61}-7}{6}}

Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{3}{7+3\cdot \xi_0}\ =\ \frac{2+\xi_0}{5+2\cdot\xi_0}\ =\ \frac{\sqrt{61}-7}{2}}

Thus for  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \xi\ge\xi_0} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{3}{7+3\cdot \xi}\ \le\ \frac{\sqrt{61}-7}{2}}

and for  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \xi\le\xi_0} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2+\xi}{5+2\cdot\xi}\ \le\ \frac{\sqrt{61}-7}{2}}

It follows that

• for one of the fractions  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{s}{t}\in\left\{\frac{a}{c},\,\frac{2\cdot a+b}{2\cdot c+d}\right\}}   the following inequality holds for every  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x\in C:}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|x-\frac{s}{t}\right|\ <\ \frac{\sqrt{61}-7}{2}}

(the choice of  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{s}{t}}   depends on  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi := \frac{d}{c}} ).

 

Divisibility

Definition  Integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a}   is divisible by integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ b}   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow:\Rightarrow\ }   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists_{c\in\mathbb{Z}}\ a=b\cdot c}

Symbolically:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b|a\ }   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow:\Rightarrow\ }   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists_{c\in\mathbb{Z}}\ a=b\cdot c}

When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a}   is divisible by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ b}   then we also say that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ b}   is a divisor of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a,}   or that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ b}   divides $\ a.$

• The only integer divisible by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 0}   is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 0}   (i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 0}   is a divisor only of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 0} ).
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\ }   is divisible by every integer.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\ }   is the only positive divisor of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 1.}
• Every integer is divisible by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 1}   (and by  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ -1} ).

 

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a|b\ \Rightarrow (-a|b\ \and\ a|\!-\!\!b)}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a|b > 0\ \Rightarrow a\le b}

 

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a|a\ }
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a|b\ \and\ b|c)\ \Rightarrow\ a|c}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a|b\ \and\ b|a)\ \Rightarrow\ |a|=|b|}

Remark  The above three properties show that the relation of divisibility is a partial order in the set of natural number  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N},}   and also in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_+} — Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 1}   is its minimal, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 0}   is its maximal element.

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a|b\ \and\ a|c)\ \Rightarrow\ (a\,|\,b\!\cdot\!d\ \and\ a\,|\,b\!+\!c\ \and\ a\,|\,b\!-\!c)}

 

Relatively prime pairs of integers

Definition  Integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a}   and $\ b$  are relatively prime   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow:\Rightarrow\ }   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 1}   is their only common positive divisor.

• Integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x}   and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 0}   are relatively prime   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftrightarrow\ |x| = 1.}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\ }   is relatively prime with every integer.
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a}   and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ b}   are relatively prime then also Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a}   and  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ -b}   are relatively prime.

Theorem 1  If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a,b,c \in \mathbb{Z}}   are such that two of them are relatively prime and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a+b=c,}   then any two of them are relatively prime.

Corollary  If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a}   and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ b}   are relatively prime then also Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a}   and  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ |a-b\,|}   are relatively prime.

Now, let's define inductively a table odd integers:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\nu_{k,n} : k\in \mathbb{Z}_+,\ 0\le n\le 2^k)}

as follows:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu_{0,0} := 0\ }     and     Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu_{0,1} :=1\ }
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu_{k+1,2\cdot n}\ :=\ \nu_{k,n}}   for  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\le n\le 2^k\ }
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu_{k+1,2\cdot n+1}\ :=\ \nu_{k,n}+\nu_{k,n+1}}   for  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\le n < 2^k\ }

for every  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ k=0,1,\dots.}

The top of this table looks as follows:

0 1

0 1 1

0 1 1 2 1

0 1 1 2 1 3 2 3 1

0 1 1 2 1 3 1 3 1 4 3 5 2 5 3 4 1

etc.

Theorem 2

• Every pair of neighboring elements of the table, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \nu_{k,n}}   and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \nu_{k,n+1}}   is relatively prime.
• For every pair of relatively prime, non-negative integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a}   and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ b}   there exist indices Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\in\mathbb{Z}_+}   and non-negative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ n<2^k}   such that:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a,b\}\ =\ \{\nu_{k,n},\nu_{k,n+1}\}}

Proof  Of course the pair

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\nu_{0,0},\nu_{0,1}\}\ =\ \{0,1\}}

is relatively prime; and the inductive proof of the first statement of Theorem 2 is now instant thanks to Theorem 1 above.

Now let $\ a$  and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ b}   be a pair of relatively prime, non-negative integers. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a+b=1}   then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \{a, b\}=\{0,1\}=\{\nu_{0,0},\nu_{0,1}\},}   and the second part of the theorem holds. Continuing this unductive proof, let's assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a+b>1.}   Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \min(a,b) > 0.}   Thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \max(a,b) < a+b}

But integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ c := \min(a,b)}   and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ d := |a-b|}   are relatively prime (see Corollary above), and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c+d\ =\ max(a,b)\ <\ a+b}

hence, by induction,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{c,d\}\ =\ \{\nu_{k,n},\nu_{k,n+1}\}}

for certain indices Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\in\mathbb{Z}_+}   and non-negative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ n<2^k.}   Furthermore:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a,b\}\ =\ \{\min(a,b), \max(a,b)\}}

It follows that one of the two options holds:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a,b\}\ =\ \{\nu_{k+1,2\cdot n},\nu_{k+1,2\cdot n+1}\}}

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a,b\}\ =\ \{\nu_{k+1,2\cdot n+1},\nu_{k+1,2\cdot n+2}\}}

End of proof

Let's note also, that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \max_{\quad 0\le n \le 2^k}\nu_{k,n}\ =\ F_{k+1}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ F_r}   is the r-th Fibonacci number.

Matrix monoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{SO}(\mathbb{Z}_+, 2)}

Definition 1  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{SO}(\mathbb{Z}_+, 2)}   is the set of all matrices

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\ :=\ \left[\begin{array}{cc}a & b\\ c & d\end{array}\right]}

such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a,b,c,d\in\mathbb{Z}_+,}   and  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \det(M) = 1,}   where  $\ \det(M):=a\cdot d-b\cdot c.$  Such matrices (and their columns and rows) will be called special.

• If

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\ :=\ \left[\begin{array}{cc}a & b\\ c & d\end{array}\right]\in\mathit{SO}(\mathbb{Z}_+, 2)}

then  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a,d > 0,}   and each of the columns and rows of M, i.e. each of the four pairs  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y) \in \{a,d\}\times \{b,c\},\ }   is relatively prime.

Obviously, the identity matrix

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{I}\ :=\ \left[\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right]}

belongs to   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{SO}(\mathbb{Z}_+, 2).}   Furthermore,  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{SO}(\mathbb{Z}_+, 2)}   is a monoid with respect to the matrix multiplication.

Example  The upper matrix and the lower matrix are defined respectively as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{U}\ :=\ \left[\begin{array}{cc}1 & 1\\ 0 & 1\end{array}\right]}   and   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}\ :=\ \left[\begin{array}{cc}1 & 0\\ 1 & 1\end{array}\right]}

Obviously Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal{U},\mathcal{L}\in \mathit{SO}(\mathbb{Z}_+,2).}   When they act on the right on a matrix M (by multipliplying M by itself), then they leave respectively the left or right column of M intact:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\cdot \mathcal{U}\ =\ \left[\begin{array}{cc}a & a+b\\ c & c+d\end{array}\right]}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\cdot \mathcal{L}\ =\ \left[\begin{array}{cc}a+b & b\\ c+d & d\end{array}\right]}

Definition 2  Vectors

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\begin{array}{c}a\\ c\end{array}\right]}     and     Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\begin{array}{c}b\\ d\end{array}\right]}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a,b,c,d\in\mathbb{Z}_+,}   are called neighbors (in that order)   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow:\Rightarrow}   matrix formed by these vectors

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\ :=\ \left[\begin{array}{cc}a & b\\ c & d\end{array}\right]}

belongs to  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{SO}(\mathbb{Z}_+, 2).}   Then the left (resp. right) column is called the left (resp. right) neighbor.

Rational representation

With every vector

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v\ :=\ \left[\begin{array}{c}a\\ c\end{array}\right]}

such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ c\ne 0,}   let's associate a rational number

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{rat}(v)\ :=\ \frac{a}{c}}

Also, let

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{rat}(v_{\infty})\ :=\ \infty}

for

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{\infty}\ :=\ \left[\begin{array}{c}1\\ 0\end{array}\right]}

Furthermore, with every matrix  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\in \mathit{SO}(\mathbb{Z}_+,2),}   let's associate the real open interval

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{span}(M)\ :=\ (\mathit{rat}(w);\mathit{rat}(v))}

and its length

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle diam(M)\ :=\ \mathit{rat}(v)-\mathit{rat}(w)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ v}   is the left, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ w}   is the right column of matrix M — observe that the rational representation of the left column of a special matrix is always greater than the rational representation of the right column of the same special matrix.

• If

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\ :=\ \left[\begin{array}{cc}a & b\\ c & d\end{array}\right]\in\mathit{SO}(\mathbb{Z}_+, 2)}

then  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ diam(M) = \frac{1}{c\cdot d}}