Title: Theory / Unimportant / Math / Number Theory / Hardy - Wright{ / 03 Farey series
Tags: examplesSwitchExample1

Name: The Farey series of order \(n\)
Definition: The series of rationals \(0\le \frac pq\le 1\) with \(p,q\) coprime, \(0<q\le n\)
Page: 28
Chapter: III
Section: 1
Variables: \(n\in\mathbb N\),
Notation: \(\mathfrak n\)
Example: for \(n=5\)

Smaller0: \(a/b\) and \(c/d\) are two consecutive terms of \(\mathfrak F_n\)
Greater0: \(bc-ad=1\)
Page: 28, 29
Chapter: III
Section: 1
Variables: \(n\in\mathbb N\),
\(a,b,c,d\in\mathbb N\)
Index: 28, 30, 31
Kind: Theo
Greater1: \(b+d>n\)
Greater2: \(\frac ab<\frac{a+c}{b+d}<\frac cd\)
Greater3: \(b\ne d\) (assuming \(n\ne 1\))

Smaller0: \(a/b\), \(c/d\) and \(e/f\) are three consecutive terms of \(\mathfrak F_n\)
Greater0: \(\frac ef=\frac {a+c}{b+d}\)
Page: 28
Chapter: III
Section: 1
Variables: \(n\in\mathbb N\),
\(a,b,c,d,e,f\in\mathbb N\)
Index: 29
Kind: Theo
Hide middle: yes

Name: A point lattice
Definition: Let \(O,P_1,P_2\) not colinear in a plane.
The set of \(O+n_1\vec{OP_1}+n_2\vec{OP_2}\) with
\(n_1,n_2\in\mathbb Z\)
Page: 32
Chapter: III
Section: 5

Name: A lattice
Definition: The lattice generated by a parallelogram
Page: 32
Chapter: III
Section: 5

Name: A lattice
Definition: The set of \((ax+by,cx+dy)\) with \(x\) or \(y\) integral
Page: 32
Chapter: III
Section: 5
Variables: \(P=(a,c),Q=(b,d)\) not colinear

Name: The fundamental lattice
Definition: The lattice such that \(O\) is the origin of a plane, \(P_1,P_2\) are the two units coordinates.
Page: 33
Chapter: III
Section: 5
Context: Number theory
Notation: \(L\)

Definition: \(ad-bc\)
Page: 33
Chapter: III
Section: 6
Variables: [$]f=\defFun{\mathbb R^2}{\mathbb R^2}{(x,y)}{ax+by,cx+dy}[/$]
Context: Number theory
Notation: \(\Delta\)

Smaller0: \(f\) is unimodular
Greater0: a bijection
Page: 34
Chapter: III
Section: 6
Variables: [$]f=\defFun{\mathbb R^2}{\mathbb R^2}{(x,y)}{ax+by,cx+dy}[/$]
Index: 32
Kind: Theo
Prefix Greater: \(f_{\mid\Lambda}\) is

Name: An unimodular transformation
Definition: A method [$]f=\defFun{\mathbb R^2}{\mathbb R^2}{(x,y)}{ax+by,cx+dy}[/$] with \(\Delta=\pm 1\)
Page: 34
Chapter: III
Section: 6

Question: The area of the parallelogram defined by \(P,Q\) is
Answer: [$]\norm{ad-bc}[/$]
Page: 34
Chapter: III
Section: 6
Variables: \(P=(a,c),Q=(b,d)\) not colinear

Smaller0: the lattice generated by \(P,Q\) is equivalent to \(L\)
Greater0: the parallelogram defined by \(OP\) and \(OQ\) is unity
Page: 34
Chapter: III
Section: 6
Variables: \(P=(a,c),Q=(b,d)\) not colinear
Index: 33
Kind: Theo

Name: A visible point
Definition: A point \(V\in\Lambda\) such that there is no \(W\in \Lambda\) such that \(O,W,V\) are aligned in this order
Page: 34
Chapter: III
Section: 6
Variables: \(P=(a,c),Q=(b,d)\) not colinear
Definition2: A point \((x,y)\in\Lambda\) such that \((x,y)=1\)

Name: Equivalent lattices
Definition: Lattices generating the same point-lattice
Page: 33
Chapter: III
Section: 5
Extra: isomorphism ?