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 if and only if there exists a real number C > 0 such that

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

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

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

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

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;

 

• $x|y\$   —   "$x\$  divides $\ y$";
• $\gcd(a,b)\$  —  the greatest common divisor of integers $\ a$  and $\ b.$

 

Divisibility

Definition  Integer $\ a$  is divisible by integer $\ b$   $\Leftarrow :\Rightarrow \$   $\exists _{c\in \mathbb {Z} }\ a=b\cdot c$

Symbolically:

$b|a\$   $\Leftarrow :\Rightarrow \$   $\exists _{c\in \mathbb {Z} }\ a=b\cdot c$

When $\ a$  is divisible by $\ b$  then we also say that $\ b$  is a divisor of $\ a,$  or that $\ b$  divides $\ a.$

• The only integer divisible by $\ 0$  is $\ 0$  (i.e. $\ 0$  is a divisor only of $\ 0$).
• $0\$  is divisible by every integer.
• $1\$  is the only positive divisor of $\ 1.$
• Every integer is divisible by $\ 1$  (and by  $\ -1$).

 

• $a|b\ \Rightarrow (-a|b\ \land \ a|\!-\!\!b)$
• $a|b>0\ \Rightarrow a\leq b$

 

• $a|a\$
• $(a|b\ \land \ b|c)\ \Rightarrow \ a|c$
• $(a|b\ \land \ b|a)\ \Rightarrow \ |a|=|b|$

Remark  The above three properties show that the relation of divisibility is a partial order in the set of natural number  $\mathbb {N} ,$  and also in $\mathbb {Z} _{+}$ — $\ 1$  is its minimal, and $\ 0$  is its maximal element.

• $(a|b\ \land \ a|c)\ \Rightarrow \ (a\,|\,b\!\cdot \!d\ \land \ a\,|\,b\!+\!c\ \land \ a\,|\,b\!-\!c)$

 

Relatively prime pairs of integers

Definition  Integers $\ a$  and $\ b$  are relatively prime   $\Leftarrow :\Rightarrow \$   $\ 1$  is their only common positive divisor.

• Integers $\ x$  and $\ 0$  are relatively prime   $\Leftrightarrow \ |x|=1.$
• $1\$  is relatively prime with every integer.

Theorem  If $\ a,b,c\in \mathbb {Z}$  are such that two of them are relatively prime and $\ a+b=c,$  then any two of them are relatively prime.