Chapter 3

Greek Number Theory

• The Role of Number Theory

• Polygonal, Prime and Perfect Numbers

• The Euclidean Algorithm

• Pell’s Equation

• The Chord and Tangent Method

• Biographical Notes: Diophantus

3.1 The Role of Number Theory• Greek mathematics

– systematic treatment of geometry (Euclid’s “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. Fermat’s Last Theorem). Solutions of many others have not been found yet (e.g. Goldbach’s conjecture)

• Nevertheless attempts to solve such problems are beneficial for the progress in mathematics

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

1 3 106

1 4 9 16

1 125 22

triangular

square

pentagonal

Results about polygonal numbers

• General formula:

Let X n,m denote mth n-agonal number. Then

X n,m = m[1+ (n-2)(m-1)/2]

• Every positive integer is the sum of four integer squares (Lagrange’s 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)

• Euler’s pentagonal theorem (1750):

2/)3(2/)3(

1 1

22

)1(1)1( kkkk

n k

kn xxx

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 > p1,p2,…, pn

– contradiction

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: 6=1+2+3, 28=1+2+4+7+14• Results:

– If 2n-1 is prime then 2n-1(2n - 1) is perfect (Euclid’s “Elements”)

– every perfect number is of Euclid’s 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?)

3.3 The Euclidean Algorithm

• Euclid’s “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

Description of the Euclidean Algorithm

1. a1 = max (a,b) – min (a,b)b1 = min (a,b)

2. (ai,bi) → (ai+1,bi+1):ai+1 = max (ai,bi) – min (ai,bi) bi+1 = min (ai,bi)

3. Algorithm terminates whenan+1 = bn+1

and thenan+1 = bn+1 = gcd (an+1,bn+1) = gcd (an,bn) = … = gcd (ai+1,bi+1) gcd (ai,bi)= …= gcd (a1,b1)= gcd (a,b)

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

3.4 Pell’s Equation• Pell’s equation is the Diophantine equation

x2 – 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 ≈ √2

Solution by Pythagoreans: recurrence relation

• x2 – 2y2 = 1

• trivial solution: x = x0 = 1, y = y0 = 0

• recurrence relation, generating larger and larger solutions:xn+1 = xn + 2yn , yn+1 = xn + yn

• then (xn)2 – 2(yn)2 = 1if n is even and (xn)2 – 2(yn)2 = -1if n is odd

How did Pythagoreans discover these recurrence relations?

• 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 = √2• Represent a and b as the sides of a rectangle

Anthyphairesis ≡ Euclidean algorithm applied to line segments and therefore to pairs of non-integers

a and b

x 1=

1

y1=√2

1

√2-11

1

√2-11

x0=√2-1

√2-1

y0=2-√2

Successive similar rectangles with sides (xn+1,yn+1) and (xn,yn)so that xn+1=xn+2yn and yn+1=xn+yn

• Note that (xn+1)2 – 2 (yn+1)2 = 0

• It turns out that the same relations generate solutions ofx2 – 2y2 = 1 or -1

• Similar procedure can be applied to 1 and √D 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 √D

• Example (cattle problem of Archimedes 287-212 BCE): x2 – 4729494y2 = 1The smallest nontrivial solution have 206,545 digits (proved in 1880)

Remarks

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 = mx+c 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, mx+c) = 0

• Note: p(x,mx+c) = 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

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 = mx+c 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, mx+c) = 0

• Indeed: p(x,mx+c) = 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)

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?

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”)