Beyond Fermat s Last Theorem
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1 Beyond Fermat s Last Theorem David Zureick-Brown Emory University UC Irvine math club a 2 + b 2 = c 2
2 Basic Problem (Solving Diophantine Equations) Analysis Let f 1,..., f m Z[x 1,..., x n ] be polynomials. Problem Describe the set { (a1,..., a n ) Z n : i, f i (a 1,..., a n ) = 0 }. Example x x 2 2 = 1 has no integer solutions David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
3 Basic Problem (Solving Diophantine Equations) Analysis Let f 1,..., f m Z[x 1,..., x n ] be polynomials. Problem Describe the set { (a1,..., a n ) Z n : i, f i (a 1,..., a n ) = 0 }. Example x x 2 2 = 1 has no integer solutions Fact Solving diophantine equations is hard. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
4 Hilbert s Tenth Problem In fact, it is provably hard. Theorem (Davis-Putnam-Robinson 1961, Matijasevič 1970) There does not exist an algorithm solving the following problem: input: f 1,..., f m Z[x 1,..., x n ]; output: YES / NO according to whether the set { (a1,..., a n ) Z n : i, f i (a 1,..., a n ) = 0 } is non-empty. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
5 Hilbert s Tenth Problem In fact, it is provably hard. Theorem (Davis-Putnam-Robinson 1961, Matijasevič 1970) There does not exist an algorithm solving the following problem: input: f 1,..., f m Z[x 1,..., x n ]; output: YES / NO according to whether the set { (a1,..., a n ) Z n : i, f i (a 1,..., a n ) = 0 } is non-empty. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
6 Hilbert s Tenth Problem In fact, it is provably hard. Theorem (Davis-Putnam-Robinson 1961, Matijasevič 1970) There does not exist an algorithm solving the following problem: input: f 1,..., f m Z[x 1,..., x n ]; output: YES / NO according to whether the set { (a1,..., a n ) Z n : i, f i (a 1,..., a n ) = 0 } is non-empty. This is still open for many other rings (e.g., R = Q). David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
7 Fermat s Last Theorem Theorem (Wiles et. al) The only solutions to the equation x n + y n = z n, n 3 are multiples of the triples (0, 0, 0), David Zureick-Brown (Emory University) (±1, 1, 0), ±(1, 0, 1), Beyond Fermat s Last Theorem (0, ±1, ±1). October 24, / 35
8 Fermat s Last Theorem Theorem (Wiles et. al) The only solutions to the equation x n + y n = z n, n 3 are multiples of the triples (±1, 1, 0), (0, 0, 0), ±(1, 0, 1), (0, ±1, ±1). This took 300 years to prove! David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
9 Fermat s Last Theorem Theorem (Wiles et. al) The only solutions to the equation x n + y n = z n, n 3 are multiples of the triples (±1, 1, 0), (0, 0, 0), ±(1, 0, 1), (0, ±1, ±1). This took 300 years to prove! David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
10 Basic Problem: f 1,..., f m Z[x 1,..., x n ] Qualitative: Does there exist a solution? Do there exist infinitely many solutions? Does the set of solutions have some extra structure (e.g., geometric structure, group structure). David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
11 Basic Problem: f 1,..., f m Z[x 1,..., x n ] Qualitative: Does there exist a solution? Do there exist infinitely many solutions? Does the set of solutions have some extra structure (e.g., geometric structure, group structure). Quantitative How many solutions are there? How large is the smallest solution? How can we explicitly find all solutions? (With proof?) David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
12 Basic Problem: f 1,..., f m Z[x 1,..., x n ] Qualitative: Does there exist a solution? Do there exist infinitely many solutions? Does the set of solutions have some extra structure (e.g., geometric structure, group structure). Quantitative How many solutions are there? How large is the smallest solution? How can we explicitly find all solutions? (With proof?) Implicit question Why do equations have (or fail to have) solutions? Why do some have many and some have none? What underlying mathematical structures control this? David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
13 Example: Pythagorean triples Lemma The equation x 2 + y 2 = z 2 has infinitely many non-zero coprime solutions. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
14 Pythagorean triples Slope = t = y x+1 x = 1 t2 1+t 2 y = 2t 1+t 2 David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
15 Pythagorean triples Lemma The solutions to are all multiples of the triples a 2 + b 2 = c 2 a = 1 t 2 b = 2t c = 1 + t 2 David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
16 The Mordell Conjecture Example The equation y 2 + x 2 = 1 has infinitely many solutions. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
17 The Mordell Conjecture Example The equation y 2 + x 2 = 1 has infinitely many solutions. Theorem (Faltings) For n 5, the equation y 2 + x n = 1 has only finitely many solutions. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
18 The Mordell Conjecture Example The equation y 2 + x 2 = 1 has infinitely many solutions. Theorem (Faltings) For n 5, the equation y 2 + x n = 1 has only finitely many solutions. Theorem (Faltings) For n 5, the equation y 2 = f (x) has only finitely many solutions if f (x) is squarefree, with degree > 4. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
19 Fermat Curves Question Why is Fermat s last theorem believable? 1 x n + y n z n = 0 looks like a surface (3 variables) 2 x n + y n 1 = 0 looks like a curve (2 variables) David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
20 Mordell Conjecture Example y 2 = (x 2 1)(x 2 2)(x 2 3) This is a cross section of a two holed torus. The genus is the number of holes. Conjecture (Mordell) A curve of genus g 2 has only finitely many rational solutions. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
21 Fermat Curves Question Why is Fermat s last theorem believable? 1 x n + y n 1 = 0 is a curve of genus (n 1)(n 2)/2. 2 Mordell implies that for fixed n > 3, the nth Fermat equation has only finitely many solutions. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
22 Fermat Curves Question What if n = 3? 1 x 3 + y 3 1 = 0 is a curve of genus (3 1)(3 2)/2 = 1. 2 We were lucky; Ax 3 + By 3 = Cz 3 can have infinitely many solutions. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
23 Congruent number problem x 2 + y 2 = z 2, xy = = 5 2, 3 4 = 2 6 David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
24 Congruent number problem x 2 + y 2 = z 2, xy = David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
25 Congruent number problem The pair of equations x 2 + y 2 = z 2, xy = has infinitely many solutions. How large Is the smallest solution? How many digits does the smallest solution have? David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
26 Congruent number problem x 2 + y 2 = z 2, xy = has infinitely many solutions. How large Is the smallest solution? How many digits does the smallest solution have? David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
27 Congruent number problem x 2 + y 2 = z 2, xy = has infinitely many solutions. How large Is the smallest solution? How many digits does the smallest solution have? x = y = z = David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
28 Congruent number problem x 2 + y 2 = z 2, xy = has infinitely many solutions. How large Is the smallest solution? How many digits does the smallest solution have? x = y = z = The denominator of z has 44 digits! David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
29 Congruent number problem x 2 + y 2 = z 2, xy = has infinitely many solutions. How large Is the smallest solution? How many digits does the smallest solution have? x = y = z = The denominator of z has 44 digits! How did anyone ever find this solution? David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
30 Congruent number problem x 2 + y 2 = z 2, xy = has infinitely many solutions. How large Is the smallest solution? How many digits does the smallest solution have? x = y = z = The denominator of z has 44 digits! How did anyone ever find this solution? Next soluton has 176 digits! David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
31 Back of the envelope calculation x 2 + y 2 = z 2, xy = Num, den(x, y, z) many, 1 min on Emory s computers. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
32 Back of the envelope calculation x 2 + y 2 = z 2, xy = Num, den(x, y, z) many, 1 min on Emory s computers. Num, den(x, y, z) many, mins = years. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
33 Back of the envelope calculation x 2 + y 2 = z 2, xy = Num, den(x, y, z) many, 1 min on Emory s computers. Num, den(x, y, z) many, mins = years many computers in the world so years David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
34 Back of the envelope calculation x 2 + y 2 = z 2, xy = Num, den(x, y, z) many, 1 min on Emory s computers. Num, den(x, y, z) many, mins = years many computers in the world so years Expected time of heat death of universe years. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
35 Fermat Surfaces Conjecture The only solutions to the equation x n + y n = z n + w n, n 5 satisfy xyzw = 0 or lie on the lines lines x = ±y, z = ±w (and permutations). David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
36 The Swinnerton-Dyer K3 surface x 4 + 2y 4 = 1 + 4z 4 David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
37 The Swinnerton-Dyer K3 surface x 4 + 2y 4 = 1 + 4z 4 Two obvious solutions (±1 : 0 : 0). David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
38 The Swinnerton-Dyer K3 surface x 4 + 2y 4 = 1 + 4z 4 Two obvious solutions (±1 : 0 : 0). The next smallest solutions are (± , ± , ± ). Problem Find another solution years to find via brute force. 2 Age of the universe ±.11 billion years (roughly ). David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
39 Fermat-like equations Theorem (Poonen, Schaefer, Stoll) The coprime integer solutions to x 2 + y 3 = z 7 are the 16 triples (±1, 1, 0), (±1, 0, 1), ±(0, 1, 1), David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
40 Fermat-like equations Theorem (Poonen, Schaefer, Stoll) The coprime integer solutions to x 2 + y 3 = z 7 are the 16 triples (±1, 1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, 2, 1), David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
41 Fermat-like equations Theorem (Poonen, Schaefer, Stoll) The coprime integer solutions to x 2 + y 3 = z 7 are the 16 triples (±1, 1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, 2, 1), (±71, 17, 2), David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
42 Fermat-like equations Theorem (Poonen, Schaefer, Stoll) The coprime integer solutions to x 2 + y 3 = z 7 are the 16 triples (±1, 1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, 2, 1), (±71, 17, 2), (± , 1414, 65), (± , 9262, 113), (± , 76271, 17). David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
43 Generalized Fermat Equations Problem What are the solutions to the equation x a + y b = z c? David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
44 Generalized Fermat Equations Problem What are the solutions to the equation x a + y b = z c? Theorem (Darmon and Granville) Fix a, b, c 2. Then the equation x a + y b = z c has only finitely many coprime integer solutions iff χ = 1 a + 1 b + 1 c 1 0. µ a µ b µ c David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
45 Known Solutions to x a + y b = z c The known solutions with are the following: 1 a + 1 b + 1 c < 1 1 p = = 3 4, = 2 9, = 71 2, = = , = = = , = David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
46 Known Solutions to x a + y b = z c The known solutions with are the following: 1 a + 1 b + 1 c < 1 1 p = = 3 4, = 2 9, = 71 2, = = , = = = , = Problem (Beal s conjecture) These are all solutions with 1 a + 1 b + 1 c 1 < 0. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
47 Generalized Fermat Equations Known Solutions Conjecture (Beal, Granville, Tijdeman-Zagier) This is a complete list of coprime non-zero solutions such that 1 p + 1 q + 1 r 1 < 0. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
48 Generalized Fermat Equations Known Solutions Conjecture (Beal, Granville, Tijdeman-Zagier) This is a complete list of coprime non-zero solutions such that 1 p + 1 q + 1 r 1 < 0. $1,000,000 prize for proof of conjecture... David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
49 Generalized Fermat Equations Known Solutions Conjecture (Beal, Granville, Tijdeman-Zagier) This is a complete list of coprime non-zero solutions such that 1 p + 1 q + 1 r 1 < 0. $1,000,000 prize for proof of conjecture......or even for a counterexample. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
50 Examples of Generalized Fermat Equations Theorem (Poonen, Schaefer, Stoll) The coprime integer solutions to x 2 + y 3 = z 7 are the 16 triples (±1, 1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, 2, 1), (±71, 17, 2), (± , 1414, 65), (± , 9262, 113), (± , 76271, 17) = 1 42 < 0 David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
51 Examples of Generalized Fermat Equations Theorem (Poonen, Schaefer, Stoll) The coprime integer solutions to x 2 + y 3 = z 7 are the 16 triples (±1, 1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, 2, 1), (±71, 17, 2), (± , 1414, 65), (± , 9262, 113), (± , 76271, 17) = 1 42 < = 0 David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
52 Examples of Generalized Fermat Equations Theorem (Darmon, Merel) Any pairwise coprime solution to the equation x n + y n = z 2, n > 4 satisfies xyz = 0. 1 n + 1 n = 2 n 1 2 < 0 David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
53 Other applications of the modular method The ideas behind the proof of FLT now permeate the study of diophantine problems. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
54 Other applications of the modular method The ideas behind the proof of FLT now permeate the study of diophantine problems. Theorem (Bugeaud, Mignotte, Siksek 2006) The only Fibonacci numbers that are perfect powers are F 0 = 0, F 1 = F 2 = 1, F 6 = 8, F 12 = 144. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
55 Examples of Generalized Fermat Equations Theorem (Klein, Zagier, Beukers, Edwards, others) The equation x 2 + y 3 = z 5 David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
56 Examples of Generalized Fermat Equations Theorem (Klein, Zagier, Beukers, Edwards, others) The equation x 2 + y 3 = z = 1 30 > 0 David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
57 Examples of Generalized Fermat Equations Theorem (Klein, Zagier, Beukers, Edwards, others) The equation x 2 + y 3 = z 5 has infinitely many coprime solutions = 1 30 > 0 David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
58 Examples of Generalized Fermat Equations Theorem (Klein, Zagier, Beukers, Edwards, others) The equation x 2 + y 3 = z 5 has infinitely many coprime solutions = 1 30 > 0 (T /2) 2 + H 3 + (f /12 3 ) 5 1 f = st(t 10 11t 5 s 5 s 10 ), 2 H = Hessian of f, 3 T = a degree 3 covariant of the dodecahedron. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
59 (p, q, r) such that χ < 0 and the solutions to x p + y q = z r have been determined. {n, n, n} Wiles,Taylor-Wiles, building on work of many others {2, n, n} Darmon-Merel, others for small n {3, n, n} Darmon-Merel, others for small n {5, 2n, 2n} Bennett (2, 4, n) Ellenberg, Bruin, Ghioca n 4 (2, n, 4) Bennett-Skinner; n 4 {2, 3, n} Poonen-Shaefer-Stoll, Bruin. 6 n 9 {2, 2l, 3} Chen, Dahmen, Siksek; primes 7 < l < 1000 with l 31 {3, 3, n} Bruin; n = 4, 5 {3, 3, l} Kraus; primes 17 l (2, 2n, 5) Chen n 3 (4, 2n, 3) Bennett-Chen n 3 (6, 2n, 2) Bennett-Chen n 3 (2, 6, n) Bennett-Chen n 3 David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
60 (p, q, r) such that χ < 0 and the solutions to x p + y q = z r have been determined. {n, n, n} Wiles,Taylor-Wiles, building on work of many others {2, n, n} Darmon-Merel, others for small n {3, n, n} Darmon-Merel, others for small n {5, 2n, 2n} Bennett (2, 4, n) Ellenberg, Bruin, Ghioca n 4 (2, n, 4) Bennett-Skinner; n 4 {2, 3, n} Poonen-Shaefer-Stoll, Bruin. 6 n 9 {2, 2l, 3} Chen, Dahmen, Siksek; primes 7 < l < 1000 with l 31 {3, 3, n} Bruin; n = 4, 5 {3, 3, l} Kraus; primes 17 l (2, 2n, 5) Chen n 3 (4, 2n, 3) Bennett-Chen n 3 (6, 2n, 2) Bennett-Chen n 3 (2, 6, n) Bennett-Chen n 3 (2, 3, 10) ZB David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
61 Faltings theorem / Mordell s conjecture Theorem (Faltings, Vojta, Bombieri) Let X be a smooth curve over Q with genus at least 2. Then X (Q) is finite. Example For g 2, y 2 = x 2g has only finitely many solutions with x, y Q. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
62 Uniformity Problem 1 Given X, compute X (Q) exactly. 2 Compute bounds on #X (Q). Conjecture (Uniformity) There exists a constant N(g) such that every smooth curve of genus g over Q has at most N(g) rational points. Theorem (Caporaso, Harris, Mazur) Lang s conjecture uniformity. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
63 Uniformity numerics g g B g (Q) (g + 1) Elkies studied K3 surfaces of the form y 2 = S(t, u, v) with lots of rational lines, such that S restricted to such a line is a perfect square. David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
64 Main Theorem (partial uniformity for curves) Theorem (Katz, Rabinoff, ZB) Let X be any curve of genus g and let r = rank Z Jac X (Q). Suppose r g 2. Then #X (Q) 84g 2 123g + 48 David Zureick-Brown (Emory University) Beyond Fermat s Last Theorem October 24, / 35
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