Chapter%203%20Greek%20Number%20Theory

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Chapter 3Greek Number Theory

• The Role of Number Theory
• Polygonal, Prime and Perfect Numbers
• The Euclidean Algorithm
• Pells Equation
• The Chord and Tangent Method
• Biographical Notes Diophantus

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3.1 The Role of Number Theory

• Greek mathematics
• systematic treatment of geometry (Euclids
  Elements
• no general methods in number theory
• Development of geometry facilitated development
  of general methods in mathematics (e.g. axiomatic
  approach)
• Number theory only a few deep results until 19th
  century (contribution made by Diophantus, Fermat,
  Euler, Lagrange and Gauss)
• Some famous problems of number theory have been
  solved recently (e.g. Fermats Last Theorem).
  Solutions of many others have not been found yet
  (e.g. Goldbachs conjecture)
• Nevertheless attempts to solve such problems are
  beneficial for the progress in mathematics

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3.2 Polygonal, Prime and Perfect Numbers

• Greeks tried to transfer geometric ideas to
  number theory
• One of such attempts led to the appearance of
  polygonal numbers

triangular
square
pentagonal
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Results about polygonal numbers

• General formulaLet X n,m denote mth n-agonal
  number. Then X n,m m1 (n-2)(m-1)/2
• Every positive integer is the sum of four integer
  squares (Lagranges Four-Square Theorem, 1770)
• Generalization (conjectured by Fermat in 1670)
  every positive integer is the sum of n n-agonal
  numbers (proved by Cauchy in 1813)
• Eulers pentagonal theorem (1750)

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Prime numbers

• An (integer) number is called prime if it has no
  rectangular representation
• Equivalently, a number p is called prime if it
  has no divisors distinct from 1 and itself
• There are infinitely many primes. Proof (Euclid,
  Elements)
• suppose we have only finite collection of primes
  p1,p2,, pn
• let p p1p2 pn 1
• p is not divisible by
• hence p is prime and p gt p1,p2,, pn
• contradiction

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Perfect numbers

• Definition (Pythagoreans) A number is called
  perfect if it is equal to the sum of its divisors
  (including 1 but not including itself)
• Examples 6123, 28124714
• Results
• If 2n-1 is prime then 2n-1(2n - 1) is perfect
  (Euclids Elements)
• every perfect number is of Euclids form (Euler,
  published in 1849)
• Open problem are there any odd perfect numbers?
• Remark primes of the form 2n-1 are called
  Mersenne primes (after Marin Mersenne
  (1588-1648))
• Open problem are there infinitely many Mersenne
  primes? (as a consequence are there infinitely
  many perfect numbers?)

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3.3 The Euclidean Algorithm

• Euclids Elements
• The algorithm might be known earlier
• Is used to find the greatest common divisor (gcd)
  of two positive integersa and b
• Applications
• Solution of linear Diophantine equation
• Proof of the Fundamental Theorem of Arithmetic

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Description of the Euclidean Algorithm

• a1 max (a,b) min (a,b)b1 min (a,b)
• (ai,bi) ? (ai1,bi1)ai1 max (ai,bi) min
  (ai,bi) bi1 min (ai,bi)
• Algorithm terminates whenan1 bn1 and
  thenan1 bn1 gcd (an1,bn1) gcd (an,bn)
  gcd (ai1,bi1) gcd (ai,bi) gcd
  (a1,b1) gcd (a,b)

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Applications

• Linear Diophantine equations
• If gcd (a,b) 1 then there are integersx and y
  such that ax by 1
• In general, there are integers x and y such that
  ax by gcd (a,b)
• Moreover, ax by d has a solution if and only
  if gcd (a,b) divides d
• The Fundamental Theorem of Arithmetic
• Lemma If p is a prime number that divides ab
  then p divides a or b
• the FTA each positive integer has a unique
  factorization into primes

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3.4 Pells Equation

• Pells equation is the Diophantine equationx2
  Dy2 1
• The best-known D. e. (after a2 b2 c2)
• Importance
• solution of it is the main step in solution of
  general quadraticD. e. in two variables
• key tool in Matiyasevich theorem on non-existence
  of the general algorithm for solving D. e.
• The simplest case x2 2y2 1 was studied by
  Pythagoreans in connection with 2 if x and y are
  large solutions then x/y v2

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Solution by Pythagoreans recurrence relation

• x2 2y2 1
• trivial solution x x0 1, y y0 0
• recurrence relation, generating larger and larger
  solutionsxn1 xn 2yn , yn1 xn yn
• then (xn)2 2(yn)2 1if n is even and
  (xn)2 2(yn)2 -1if n is odd

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How did Pythagoreans discover these recurrence
relations?
Anthyphairesis Euclidean algorithm applied to
line segments and therefore to pairs of
non-integersa and b

• When the ratio a/b is rational the algorithm
  terminates
• If a/b is irrational it continues forever
• Apply this algorithm to a 1 and b v2
• Represent a and b as the sides of a rectangle

Successive similar rectangles with sides
(xn1,yn1) and (xn,yn) so that xn1xn2yn and
yn1xnyn
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Remarks

• Note that (xn1)2 2 (yn1)2 0
• It turns out that the same relations generate
  solutions ofx2 2y2 1 or -1
• Similar procedure can be applied to 1 and vD to
  obtain solutions of x2 Dy2 1 (Indian
  mathematician Brahmagupta, 7th century CE)
• To obtain recurrence we need the recurrence of
  similar rectangles (proved by Lagrange in 1768)
• Continued fraction representation for vD
• Example (cattle problem of Archimedes 287-212
  BCE) x2 4729494y2 1The smallest nontrivial
  solution have 206,545 digits (proved in 1880)

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3.5 The Chord and Tangent Method

• Generalization of Diophantus method to find all
  rational points on the circle
• Consider any 2nd degree algebraic curve p(x,y)
  0 where p is a quadratic polynomial (in two
  variables) with integer coefficients
• Consider rational point x r1, y s1 such that
  p(r1,s1) 0
• Consider a line y mxc with rational slope m
  through (r1,s1) (chord)
• This line intersects curve in the second point
  which is the second solution of equation p (x,
  mxc) 0
• Note p(x,mxc) k(x-r1)(x-r2) 0
• Thus we obtain the second rational point
  (r2,s2)(where s2 mr1 c)
• All rational points on 2nd degree curve can be
  obtained in this way

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If p(x,y) has degree 3

• Consider an algebraic curve p(x,y) 0 of degree
  3
• Consider base rational point x r1, y s1 such
  that p(r1,s1) 0
• Consider a line y mxc through (r1,s1) which is
  tangent to p(x,y) 0 at (r1,s1)
• It has rational slope m
• This line intersects curve in the second point
  which is the third solution of the equation p (x,
  mxc) 0
• Indeed p(x,mxc) k(x-r1)2(x-r2) 0 (r1 is a
  double root)
• Thus we obtain the second rational point (r2,s2)
  (where s2 mr1 c), and so on
• This tangent method is due to Diophantus and was
  understood by Fermat and Newton (17th century)

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Does this method give us all rational points on a
cubic?

• In general, the answer is negative
• The slope is no longer arbitrary
• Theorem (conjectured by Poincaré (1901), proved
  by Mordell (1922)) All rational points can be
  generated by tangent and chord constructions
  applied to finitely many points
• Open problem is there an algorithm to find this
  finite set of such rational points on each cubic
  curve?

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2.6 Biographical Notes Diophantus

• Approximately between 150 and 350 CE
• Lived in Alexandria
• Greek mathematics was in decline
• The burning of the great library in Alexandria
  (640 CE) destroyed all details of Diophantus
  life
• Only parts of Diophantus work survived (e.g.
  Arithmetic)