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 19 th 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 triangular square pentagonal

Results about polygonal numbers General formula: Let X n,m denote m th 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):

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: p 1,p 2,…, p n –let p = p 1 p 2 … p n +1 –p is not divisible by –hence p is prime and p > p 1,p 2,…, p n –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= Results: –If 2 n -1 is prime then 2 n-1 (2 n - 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 2 n -1 are called Mersenne primes (after Marin Mersenne ( )) 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 integers a and b Applications: –Solution of linear Diophantine equation –Proof of the Fundamental Theorem of Arithmetic

Description of the Euclidean Algorithm 1.a 1 = max (a,b) – min (a,b) b 1 = min (a,b) 2.(a i,b i ) → (a i+1,b i+1 ): a i+1 = max (a i,b i ) – min (a i,b i ) b i+1 = min (a i,b i ) 3.Algorithm terminates when a n+1 = b n+1 and then a n+1 = b n+1 = gcd (a n+1,b n+1 ) = gcd (a n,b n ) = … = gcd (a i+1,b i+1 ) gcd (a i,b i )= …= gcd (a 1,b 1 )= gcd (a,b)

Applications Linear Diophantine equations –If gcd (a,b) = 1 then there are integers x 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 x 2 – Dy 2 = 1 The best-known D. e. (after a 2 + b 2 = c 2 ) Importance: –solution of it is the main step in solution of general quadratic D. e. in two variables –key tool in Matiyasevich theorem on non-existence of the general algorithm for solving D. e. The simplest case x 2 – 2y 2 = 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 x 2 – 2y 2 = 1 trivial solution: x = x 0 = 1, y = y 0 = 0 recurrence relation, generating larger and larger solutions: x n+1 = x n + 2y n, y n+1 = x n + y n then (x n ) 2 – 2(y n ) 2 = 1if n is even and (x n ) 2 – 2(y n ) 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 y 1= √2 1 √ x 0 =√2-1 √2-1 y 0 =2-√2 Successive similar rectangles with sides (x n+1,y n+1 ) and (x n,y n ) so that x n+1 =x n +2y n and y n+1 =x n +y n

Note that (x n+1 ) 2 – 2 (y n+1 ) 2 = 0 It turns out that the same relations generate solutions of x 2 – 2y 2 = 1 or -1 Similar procedure can be applied to 1 and √D to obtain solutions of x 2 – Dy 2 = 1 (Indian mathematician Brahmagupta, 7 th 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 BCE ): x 2 – y 2 = 1 The 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 2 nd degree algebraic curve: p(x,y) = 0 where p is a quadratic polynomial (in two variables) with integer coefficients Consider rational point x = r 1, y = s 1 such that p(r 1,s 1 ) = 0 Consider a line y = mx+c with rational slope m through (r 1,s 1 ) (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-r 1 )(x-r 2 ) = 0 Thus we obtain the second rational point (r 2,s 2 ) (where s 2 = mr 1 + c) All rational points on 2 nd 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 = r 1, y = s 1 such that p(r 1,s 1 ) = 0 Consider a line y = mx+c through (r 1,s 1 ) which is tangent to p(x,y) = 0 at (r 1,s 1 ) 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-r 1 ) 2 (x-r 2 ) = 0 (r 1 is a double root) Thus we obtain the second rational point (r 2,s 2 ) (where s 2 = mr 1 + c), and so on This tangent method is due to Diophantus and was understood by Fermat and Newton (17 th 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”)