What is number theory and why is it important? Science Talks About - PDF document

What is number theory and why is it important? Science Talks About Research for Staff (STARS) lecture Henri Darmon McGill University Montreal March 13 2007 http://www.math.mcgill.ca/darmon /slides/slides.html

What is number theory? What is a number? Counting numbers (ca. 30000 BC)  A pair of shoes ,   A brace of geese , = the number two . ⇒ A married couple .   1 , 2 , 3 , . . . , 4265173 , . . . The number zero : (cd 960 AD). Integers : . . . , − 2 , − 1 , 0 , 1 , 2 , 3 , . . . Fractions : 2 3 , 1 2 , − 5 7 , . . . 1

Numbers as lengths The number line −1 0 1 2 −2 1/2 Pythagorean creed : “All is number”. The Pythagorean Theorem. c 2 2 2 a + b = c a b 2

The square root of two c 1 1 1 2 + 1 2 = c 2 , hence c 2 = 2 . √ c = 2 , but... 3

A troubling discovery Hippasus of Metapontum (ca 500 BC): √ 2 cannot be expressed as a fraction! √ 2 = 1 . 4142135623730950488016887 . . . 2 / 7 = 0 . 285714285714285714285714285714 . . . According to legend, Hippasus made this trou- bling discovery on a boat bound for Samos. He was thrown overboard by the fanatical dis- ciples of Pythagoras, in an attempt to conceal his discovery. 4

√ Constructing 2 √ To construct 2, one needs to be able to con- struct an accurate right angle . Practical applications: architecture, geodesy. Natural approach: find right-angled triangles with integer side lengths. I.e., solve the equation a 2 + b 2 = c 2 in integers. 5

Pythagorean triangles 5 4 3 There are infinitely many distinct Pythagorean triangles. c = u 2 + v 2 . a = u 2 − v 2 , b = 2 uv, This is one of the very early results of number theory. 6

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Congruent numbers An integer n is a congruent number if it is the area of a right-angled triangle with rational side lengths (a Pythagorean triangle). Examples : 6 is a congruent number... 5 4 3 8

... and so is 157! 224403517704336969924557513090674863160948472041 8912332268928859588025535178967163570016480830 6803298587826435051217540 411340519227716149383203 21666555693714761309610 411340519227716149383203 9

The equation E n Elementary manipulations show: Fact n is a congruent number if and only if the equation E n : y 2 = x 3 − n 2 x has a non-zero solution (with y � = 0). Question : Study the set of rational solutions to the equation E n . One is thus led to a question about whether a cubic equation has rational solutions. 10

Elliptic Curves Definition : An elliptic curve is an equation of the form y 2 = x 3 + ax + b, with ∆ := 4 a 3 − 27 b 2 � = 0 . Why elliptic curves? 1. They arise naturally in studying congruent numbers. 2. They are relatively simple equations of not too large degree. 3. They are endowed with a remarkably rich mathematical structure. 11

The duplication rule The duplication law allows us to produce new solutions from old ones. y 2 3 y = x + a x + b R Q x P P+Q The duplication law on an elliptic curve 12

Fermat Theorem (Fermat) The elliptic curve y 2 = x 3 − x has no non-zero solution (i.e., 1 is not a con- gruent number). Fermat’s descent : Most importantly, Fermat introduced a general approach, for checking (in some cases) whether an elliptic curve has a rational solution or not. In general, studying the rational solutions of an elliptic curve equation can get quite com- plicated! 13

An example of Bremner and Cassels The solutions of the equation E : y 2 = x 3 + 877 x can be obtained from the basic solution x = 612776083187947368101 2 78841535860683900210 2 25625626798892680938877 68340455130896486691 y = 53204356603464786949 78841535860683900210 3 by repeated application of the duplication rule. Finding a “basic solution” can be a daunting task! 14

The main question Given n , find a criterion for E n to have a non- zero rational solution. Modern approach: given a prime p , define � � 1 ≤ x ≤ p 1 ≤ y ≤ p p divides y 2 − ( x 3 − n 2 x ) N p = # . Basic idea: if E n has many rational solutions, the numbers N p should have a tendency of be- ing large. Fix a large prime P and consider L ( P ) = N 2 2 × N 3 3 × N 5 5 × N 7 7 × N 11 11 × · · · × N P P . Wiles : The N p satisfy an “explicit pattern” which can be used to “analyze” L ( P ). This undersanding forms the basis of Wiles’ proof of “Fermat’s Last Theorem”. 15

A Theorem of Coates-Wiles Theorem . If L ( P ) remains bounded as P gets large, then E n has no non-zero solution, and hence n is not a congruent number. Remarks : 1. The condition that L ( P ) remains bounded is not hard to check numerically. 2. This result has been extended to all elliptic curves by Gross-Zagier-Kolyvagin. 16

The Birch and Swinnerton-Dyer Conjecture It states (in a special case): Conjecture . If L ( P ) is unbounded as P grows, then E n has infinitely many rational solutions (and hence, n is a congruent number.) We are still far from being able to prove this! Difficulty . Number theory disposes of a very limited arsenal of methods for producing solu- tions to equations like E n (as opposed to show- ing they do not exist.) Understanding the mysterious process whereby the size of N p forces the presence of a rational solutions of E n would be a great step forward. 17

The Millenium Prize Problems The Clay Mathematics Institute in Cambridge, Mass has offered a 1,000,000$ prize for the solution of any of the following problems 1. The Birch and Swinnerton-Dyer conjecture. 2. The Hodge Conjecture 3. Solution of Navier-Stokes Equations 4. The P vs NP problem 5. The Poincar´ e Conjecture 6. The Riemann Hypothesis 7. Yang-Mills Theory The 5th has recently been solved (by Grig- ori Perelman, a Russian mathematician.) The others are still open! 18

Why is number theory important? A utilitarian defense Number theory is interested in highly struc- tured objects, like elliptic curves. Such objects have found widespread real-world applications , in cryptography, coding theory, etc... Data encryption an compression rely on fun- damental ideas from number theory. 19

Why is number theory important? The defense of Gustav Jacobi “ Monsieur Fourier avait l’opinion que le but principal des math´ ematiques ´ etait l’utilit´ e publique et l’explication des ph´ enom` enes naturels. Un philosophe tel que lui aurait dˆ u savoir que le but unique de la Science, c’est l’honneur de l’esprit humain et que, sous ce titre, une ques- tion de nombres vaut bien une question de syst` eme du monde.” 20

Conclusion Whether it strives for safer internet shopping or the elevation of the human spirit, Number Theory is subject which is rich and fascinating, and replete with many tantalising mysteries. 21