Ancient Mathematical Techniques: Euclidean Algorithm, Chinese Remainder Theorem, Pell’s Equation

1. Chapter 4Number Theory in Asia • The Euclidean Algorithm • The Chinese Remainder Theorem • Linear Diophantine Equations • Pell’s Equation in Brahmagupta • Pell’s Equation in Bhâskara II • Rational Triangles • Biographical Notes: Brahmagupta and Bhâskara

2. 5.1 The Euclidean Algorithm • Major results known in ancient China and India (independently) • Pythagorean theorem and triples • Concept of π • Euclidean algorithm (China, Han dynasty,200 BCE – 200 CE) • Practical applications of Euclidean algorithm • Chinese remainder theorem • India: solutions of linear Diophantine equations and Pell’s equation

3. 5.2 The Chinese Remainder Theorem • Example: find a number that leaves remainder 2 on division by 3, rem. 3 on div. by 5, and rem. 2 on div. by 7 • In terms of congruencies:x ≡ 2 mod 3, x ≡ 3 mod 5, x ≡ 2 mod 7 • Solution: x = 23 • “General method” - Mathematical Manual by Sun Zi(late 3rd century CE): • If we count by threes and there is a remainder 2, put down 140 • If we count by fives and there is a remainder 3, put down 63 • If we count by seven and there is a remainder 2, put down 30 • Add them to obtain 233 and subtract 210 to get the answer

4. Explanation • 140 = 4 x (5 x 7) leaves remainder 2 on division by 3 and remainder 0 on division by 5 and 7 • 63 = 3 x (3 x 7) leaves remainder 3 on division by 5 and remainder 0 on division by 3 and 7 • 30 = 2 x (3 x 5) leaves remainder 2 on division by 3 and remainder 0 on division by 5 and 7 • Therefore their sum 233 leaves remainders 2, 5, and 2 on division by 3, 5, and 7, respectively • Subtract integral multiple of 3 x 5 x 7 = 105 to obtain the smallest solution:233 – 2 x 105 = 233 – 210 = 23 • Question: Why do we choose 140, 63 and 30 (or, more precisely 4, 3 and 2)?

5. Sun Zi: • If we count by threes and there is a remainder 1, put down 70 • If we count by fives and there is a remainder 1, put down 21 • If we count by seven and there is a remainder 1, put down 15 • We have: • 70 = 2 x (5 x 7) – smallest multiple of 5 and 7 leaving remainder 1 on division by 3 • Multiply it by 2 to get remainder 2 on division by 3:140 = 2 x 70 = 2 x 2 (5 x 7) = 4 (5 x 7)

6. Inverses modulo p • Definitionb is called inverse of a modulo p if ab ≡ 1 mod p • Examples: • 2 is inverse of 3 modulo 5 • 3 is inverse of 3 modulo 8 • 2 is inverse of 35 modulo 3 • Does inverse of a modulo p exist ? • Example: does inverse of 4 modulo 6 exist? • If we had 4b ≡ 1 mod 6 then 4b – 1 were divisible by 6 and therefore 1 were divisible by 2 = gcd (4,6), which is impossible!

7. General method of finding inverses • Qin Jiushao, 1247 - used Euclidean algorithm to find inverses • Suppose ax = 1 mod p where x is unknown • Then ax – 1 = py ↔ ax - py = 1 • This equation has solutions if and only ifgcd (a,p) = 1 and in this case solutions can be found from Euclidean algorithm! • In particular, if p is prime then inverse always exist

8. Chinese remainder theorem • p1, p2, … pk - relatively prime integers, i.e. gcd (pi, pj) =1 for all i ≠ j • remainders: r1, r2, …, rk such that 0 ≤ri< pi • Then there exists integer n satisfying the system of k congruencies:n ≡ r1 mod p1n ≡ r2 mod p2.....................n ≡ rk mod pk

9. 5.3 Linear Diophantine Equations • ax + by = c • Euclidean algorithm: • China: between 3rd century CE and 1247 (Qin Jiushao) • India: Âryabhata (499 CE) • Bhaskara I (India, 522) reduced problem to finding a’x + b’y = 1 where a’ = a / gcd (a,b) and b’ = b / gcd (a,b) • Criterion for an integer solution:Equation ax + by = c has an integer solution if and only if gcd (a,b) divides c

10. 5.4 Pell’s Equation in Brahmagupta • China: development of algebra and approximate methods, integer solutions for linear equations, but not integer solutions for nonlinear equations • India: less progress in algebra but success in finding solutions of Pell’s equation:Brahmagupta “Brâhma-sphuta-siddhânta” 628 CE

11. Brahmagupta’s method • Pell’s equation: x2 – Ny2 = 1 • Method is based on Brahmagupta's discovery of identity: • Note: if we let N = -1 we obtain identity discovered by Diophantus:

12. Composition of triples • Consider equations(1) x2 – Ny2 = k1 and (2) x2 – Ny2 = k2 • x1, y1 – sol. of (1), x2, y2 – sol. of (2) • Then the identity implies thatx =x1x2+Ny1y2 and y =x1y2+x2y1 is a solution of x2 – Ny2 = k1 k2 • We therefore define composition of triples(x1, y1, k1 ) and (x2, y2, k2 ) equal to the triple(x1x2+Ny1y2, x1y2+x2y1, k1 k2 ) • Thus if k1= k2 =1 one can obtain arbitrary large solutions ofx2 – Ny2 = 1 (e.g. start from some obvious solution and compose it with itself)

13. Moreover… • It turns out that we can obtain solutions of x2 – Ny2 = 1 composing solutions of x2 – Ny2 = k1 and x2 – Ny2 = k2 even when k1,k2 > 1 • Indeed: composing (x1, y1, k1 ) with itself gives integer solution of x2 – Ny2 = (k1 )2 and hence rational solution of x2 – Ny2 = 1 • Example (Brahmagupta: “a person solving this problem within a year is a mathematician”)x2 – 92y2 = 1 • Consider “auxiliary” equation x2 – 92y2 = 8 • It has obvious solution (10, 1, 8) • Composing it with itself we get (192, 20, 64) which is a solution of x2 – 92y2 = 82 • Dividing both sides by 82 we obtain (24, 5/2, 1) which is a rational solution of x2 – 92y2 = 1 • Composing it with itself we get (1151, 120, 1) which means thatx = 1151, y = 120 is a solution of x2 – 92y2 = 1

14. 5.5 Pell’s Equation in Bhâskara II • Brahmagupta: • invented composition of triples • proved that if x2 – Ny2 = k has an integer solution for k = 1, 2 ,4 then x2 – Ny2 = 1 has integer solution • Bhâskara: • first general method for solving the Pell equation (“Bîjaganita” 1150 CE) • method is based on Brahmagupta’s approach

15. Method of Bhâskara • Goal: find a non-trivial integer solution of x2 – Ny2 = 1 • Let a and b are relatively prime such that a2 – Nb2 = k • Consider trivial “equation” m2 – N x 12 = m2 – N • Compose triples (a, b, k) and (m, 1, m2-N) • We get (am + Nb, a+bm, k (m2 – N) ) • Dividing by k we get: ((am + Nb) / k, (a+bm) / k, (m2 – N) ) / k • Choose m so that (a+bm) / k = b1is an integer AND so thatm2 – N is as small as possible • It turns out that (am + Nb) / k = a1and (m2-N) / k = k1 are integers • Now we have (a1)2 – N (b1)2 = k1 • Repeat the same procedure to obtain k2and so on • The goal is to get ki = 1, 2 or 4 and use Brahmagupta’s method

16. Example • Consider x2 – 61y2 = 1 • Equation 82 – 61 x 12 = 3 gives (a, b, k) = (8, 1, 3) • Composing (8, 1, 3) with (m,1, m2 – 61) we get(8m + 61, 8+m, 3(m2 – 61)) • Dividing by 3 we get ( (8m + 61) / 3, (8+m) / 3, m2 – 6) • Letting m = 7 we get (39, 5, - 4) • (Brahmagupta) Dividing by 2 (since 4 = 22) we get(39/2, 5/2, -1) • Composing it with itself we get (1523 / 2, 195 /2, 1) • Composing it with (39/2, 5/2, -1) we get (29718, 3805, -1) • Composing it with itself we get (1766319049, 226153980, 1) which is a solution of x2 – 61y2 = 1 ! • In fact, it is the minimal nontrivial solution!

17. 5.6 Rational Triangles • Definition A triangle is called rational if it has rational sides and rational area • Equivalently: rational sides and altitudes • Brahmagupta’s Theorem:Parameterization of rational trianglesIf a, b, c are sides of a rational triangle then for some rational numbers u, v and w we have:a = u2 / v + v, b = u2 / w + wc = u2 / v – v + u2 / w – w

18. Stronger Form • Any rational triangle is of the forma = (u2 + v2) / v, b = (u2 + w2) / wc = (u2 – v2 ) / v + (u2 –w2 ) / wfor some rational numbers u, v, w with the altitude h = 2usplitting side c into segments c1 = (u2 – v2 ) / v and c2 = (u2 – v2 ) / v

19. 5.7 Biographical Notes:Brahmagupta and Bhâskara II • Brahmagupta (598 – (approx.) 665 CE) • “Brâhma-sphuta-siddhânta” • teacher from Bhillamâla (now Bhinmal, India) • prominent in astronomy and mathematics • Pell’s equation • general solution of quadratic equation • area of a cyclic quadrilateral (which generalizes Heron’s formula for the area of triangle) • parameterization of rational triangles

20. Bhâskara II (1114 – 1185) • greatest astronomer and mathematician in 12th-century India • head of the observatory at Ujjain • “Līlāvatī” (work named after his daughter)