Chapter 3 Greek Number Theory

1. Chapter 3Greek 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

2. 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. 6 10 3 1 1 4 9 16 1 5 12 22 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

4. Results about polygonal numbers • General formula:Let X n,mdenote 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):

5. Prime numbers • An integer 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 p1,p2,…, pn • hence p is prime and p > p1,p2,…, pn • contradiction

6. 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 even 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-1are 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?)

7. 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

8. Description of the Euclidean Algorithm • a1 = max (a,b) – min (a,b)b1 = min (a,b) • (ai,bi) → (ai+1,bi+1):ai+1 = max (ai,bi) – min (ai,bi) bi+1 = min (ai,bi) • 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)

9. 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

10. 3.4 Pell’s Equation • Pell’s 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 ≈ √2

11. 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

12. y1=√2 1 √2-1 1 √2-1 √2-1 x1=1 1 1 y0=2-√2 x0=√2-1 Anthyphairesis≡ Euclidean algorithm applied to line segments and therefore to pairs of non-integersa and b 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 Successive similar rectangles with sides (xn+1,yn+1) and (xn,yn) so that xn+1=xn+2yn and yn+1=xn+yn

13. Remarks • 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)

14. 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

15. 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)

16. 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?

17. 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”)