Number Theory Revival: From Diophantus to Fermat

1. Chapter 11The Number Theory Revival • Between Diophantus and Fermat • Fermat’s Little Theorem • Fermat’s Last Theorem • Rational Right-Angled Triangles • Rational points on cubics of genus 0 and 1

2. 11.1 Between Diophantus and Fermat • China, Middle Ages (11th -13th centuries) • Chinese Remainder Theorem • Pascal’s triangle • Levi ben Gershon (1321) • “combinatorics and mathematical induction” (formulas for permutations and combinations) • Blaise Pascal (1654) • unified the algebraic and combinatorial approaches to “Pascal’s triangle”

3. Pascal’s trianglein Chinese mathematics The Chineseused Pascal’s triangle to find the coefficients of (a+b)n

4. Pascal’s Triangle

5. As we know now, the kth element of nth row is since • Thus Pascal’s triangle expresses the following property of binomial coefficients:

6. Letting k = j + 1 • in the second • sum we get • Indeed, suppose that for all n we have • Then • Knowing that C0n-1=1, Cn-1n-1 =1 and replacing j by k in the • first sum we obtain

7. Combinations, permutations,and mathematical induction • Levi ben Gershon (1321) gave the formula for the number of combinations of n things taken k at a time: • He also pointed out that the number of permutations of n elements is n! • The method he used to show these formulas is very close tomathematical induction

8. Why “Pascal’s Triangle” ? • Pascal demonstrated (1654) that the elements of this triangle can be interpreted in two ways: • algebraically as binomial coefficients • combinatorially as the number of combinations of n things taken k at a time • As application he solved problem of division of stakes and founded the mathematical theory of probabilities

9. 11.2 Fermat’s Little Theorem • Theorem (Fermat, 1640)If p is prime and n is relatively prime to p (i.e. gcd (n,p)=1) then np – 1≡ 1 mod p • Equivalently, np-1 – 1 is divisible by p if gcd (n,p)=1 ornp – n is divisible by p (always) • Note: Fermat’s Little Theorem turned out to be very important for practical applications – it is an important part in the design of RSA encryption algorithm as well as primality testing • Fermat was interested in the expressions of the form2m – 1 (in connection with perfect numbers) and, at the same time, he was investigating binomial coefficients • Fermat’s original proof of the theorem is unknown

10. Proof • Proof can be conducted in two alternative ways: • iterated use of binomial theorem • application of the following multinomial theorem:

11. 11.3 Fermat’s Last Theorem “On the other hand, it is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as a sum of two fourth powers or, in general, for any number which is a power higher than second to be written as a sum of two like powers. I have a truly marvellous demonstration of this proposition which this margin is too small to contain.” written by Fermat in the margin of his copy of Bachet’s translation of Diophantus’ “Arithmetica”

12. Theorem There are no triples (a,b,c) of positive integers such thatan + bn = cn where n > 2 is an integer • Proofs for special cases: • Fermat for n = 4 • Euler for n = 3 • Legendre and Dirichlet for n = 5, • Lame for n = 7 • Kummer for all prime n < 100 except 37, 59, 67 • Note: it is sufficient to prove theorem for all prime exponents (except 2) and for n = 4, since if n = mp where p is prime and an + bn = cn then(am)p + (bm)p = (cm)p

13. First significant step (after Kummer):Proof of Mordell’s conjecture (1922) about algebraic curves given by Falting (1983) • Applied to the “Fermat curve” xn + yn = 1 for n > 3, this conjecture provides the following statement • Fermat curve contains at most finitely manyof rational points for each n > 3 • Therefore, Falting’s result imply that equation an + bn = cn can have at most finitely many solutionsfor each n > 3 • The complete proof of Fermat’s Last Theorem is due to Andrew Wiles and follows from much more general statement (first announcement in 1993, gap found, filled in 1994, complete proof published in 1995)

14. 11.4 Rational Right-Angled Triangles • What can be the area of a right-angled triangle with rational sides? • Fermat (answering a question of Bachet):it cannot be a square (of an integer) • Method of ”infinite descent” • Also contains a proof of Fermat’s last theorem for n=4

15. Proof • We may assume (using scaling) that the sides of the triangle are relatively prime integers • Thus they form a primitive Pythagorean triple:m2+n2, 2mn, m2-n2 • Claim: m, n, m+n, m-n are pairwise relatively prime • Claim: m+n, m-n are odd • The area is mn(m+n)(m-n) • If it is a square, all factors must be squares as well • From this we can obtain a smaller triangle with integer sides whose area is also a square, and so on • This is the method of “infinite descent”

16. 11.5 Rational Points on Cubic of Genus 0 • Algebraic curve: p(x,y)=0, p - polynomial of degree d • Singular point on a curve is a point (a,b) where px(a,b)=py(a,b)=p(a,b)=0 • There is a way to assign, to each singularity, a non-negative integer s, that describes complexity of a singularity • Define the genus of the curve by(d-1)(d-2)/2 – (sum of all s over all singularities)

17. Example: Folium of Descartes • x3+y3=3xy (d = 3) • The only singular point is (0,0) • It is a point of multiplicity 2 • Genus of the folium is(3-1)(3-2)/2 – 1 =0 • In general, genus of a cubic curve is either 0 or 1 (the latter occurs when there is no singular points)

18. Example: Folium of Descartes • x3+y3=3xy

19. Example: Folium of Descartes • x3+y3=3xy (d = 3) • The only singular point is (0,0) • It is a point of multiplicity 2 • Genus of the folium is(3-1)(3-2)/2 – 1 =0 • In general, genus of a cubic curve is either 0 or 1 (the latter occurs when there is no singular points)

20. Y R Q X 1 O -1 Recall: rational points on second-order curves • Base point (trivial solution) Q(x,y) = (-1,0) • Line through Q with rational slope ty = t(x+1)intersects the circle at a second rational point R • As t varies we obtain all rational points on the circle which have the formx = (1-t2) / (1+t2), y = 2t / (1+t2)where t = p/q • This is a parameterization using rational functions!

21. Parameterization of cubic curves • It can be shown that the cubic curves of genus 0 are precisely those that can be parameterized by rational functions • For folium: x=3t/(1+t3), y=3t2/(1+t3) • Moreover, the secant line through the double point of the folium intersects it in another rational point, and all rational points on the folium can be obtained in this way

22. 11.6 Rational Points on Cubic of Genus 1 • Recall: genus=(d-1)(d-2)/2 – (sum of all s over all singularities) • So for cubic curves we get (3-1)(3-2)/2 - K • Thus the genus of a cubic is 1 if and only if it has no singular points • Thus, a cubic of genus 1 has no double points or cusps, and cannot be parameterized by rational functions • Note: it can be shown that all cubic curves have either genus 0 or genus 1

23. Parameterization in case of Genus 1 • Newton represented any cubic in one of the following form (using suitable choice of coordinates):axy2+by=cx3+dx2+ex+fxy=ax3+bx2+cx+dy2 = ax3+bx2+cx+dy=ax3+bx2+cx+d

24. Parameterization in case of Genus 1 • Consider a curvey2 = ax3+bx2+cx+d • What functions can be used to parameterize such curve? • Abel, Gauss, and Jacobi defined (independently) a special class of functions called elliptic functions (19th century) • Clebsch used them to parameterize cubic curves

25. Elliptic curves • y2 = x3 + ax +b

26. Examples of elliptic curves 5 y y 4 4 3 3 2 2 1 1 0 0 -1 2 3 -1 2 1 3 1 -2 -3 -2 x x -1 -1 -2 -2 -3 -3 -4 -4 -5 y2 = x3 – 2x y2 = x3 – x + 1

27. Elliptic curves • y2 = x3 + ax +b • We can consider a secant line intersecting the curve at three points • If the coefficients a, b are rational and if two of the intersection points are rational, the third is also rational

28. Addition of points on an elliptic curve • We can add points on elliptic curve Е as follows: • Consider P andQ on E • Draw the line PQ and find its intersection T with E • S that is symmetric to T with respect to the y-axis, is the sum of P and Q, S = P+Q

29. Addition in coordinates Q(xq,yq) P(xp,yp) P+Q(x,y) y2 = x3 + ax +b

30. Addition on elliptic curves • The formulas lead to addition theorem for elliptic functions (Fagnano (1718), Euler (1784), discovered analyzing the integrals that are used in the definition of elliptic functions). • Modern use: elliptic curve cryptography

31. Pierre de Fermat Born: 1601 in Beaumont (near Toulouse, France) Died: 1665 in Castres (France)

32. Wealthy family • University of Orléans:1623 – 1626, bachelor in civil law • First studies in mathematics: 1629 • Began legal work around 1630-31 • Mathematics was a hobby • Did not publish his results (they appear in letters, notes, etc.) • First work in analytic geometry • Calculus (maxima, minima, tangents) • Number theory and algebraic geometry