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, A married couple. = ⇒ the number two. 1, 2, 3, . . . , 4265173, . . . The number zero: (cd 960 AD). Integers: . . . , −2, −1, 0, 1, 2, 3, . . . Fractions: 2

3, 1 2, −5 7, . . .

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Numbers as lengths

The number line

−1 −2 1 2

1/2

Pythagorean creed: “All is number”. The Pythagorean Theorem.

c b a

a + b = c

2 2 2 2

The square root of two

1 1 c

12 + 12 = c2, hence c2 = 2. c = √ 2, but...

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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.

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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 a2 + b2 = c2 in integers.

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Pythagorean triangles

3 4 5

There are infinitely many distinct Pythagorean triangles. a = u2 − v2, b = 2uv, c = u2 + v2. This is one of the very early results of number theory.

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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...

3 4 5

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... and so is 157!

6803298587826435051217540 411340519227716149383203 21666555693714761309610 411340519227716149383203 224403517704336969924557513090674863160948472041 8912332268928859588025535178967163570016480830

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The equation En

Elementary manipulations show: Fact n is a congruent number if and only if the equation En : y2 = x3 − n2x has a non-zero solution (with y = 0). Question: Study the set of rational solutions to the equation En. One is thus led to a question about whether a cubic equation has rational solutions.

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Elliptic Curves

Definition: An elliptic curve is an equation of the form y2 = x3 + ax + b, with ∆ := 4a3 − 27b2 = 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.

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The duplication rule

The duplication law allows us to produce new solutions from old ones.

x y y = x + a x + b

2 3

P Q R P+Q

The duplication law on an elliptic curve

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Fermat

Theorem (Fermat) The elliptic curve y2 = x3 − 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!

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An example of Bremner and Cassels

The solutions of the equation E : y2 = x3 + 877x can be obtained from the basic solution x = 6127760831879473681012 788415358606839002102 y = 25625626798892680938877 68340455130896486691 53204356603464786949 788415358606839002103 by repeated application of the duplication rule. Finding a “basic solution” can be a daunting task!

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The main question

Given n, find a criterion for En to have a non- zero rational solution. Modern approach: given a prime p, define Np = #

• 1 ≤ x ≤ p

1 ≤ y ≤ p p divides y2 − (x3 − n2x)

• .

Basic idea: if En has many rational solutions, the numbers Np should have a tendency of be- ing large. Fix a large prime P and consider L(P) = N2 2 × N3 3 × N5 5 × N7 7 × N11 11 × · · · × NP P . Wiles: The Np 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”.

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A Theorem of Coates-Wiles

• Theorem. If L(P) remains bounded as P gets

large, then En 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.

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The Birch and Swinnerton-Dyer Conjecture

It states (in a special case):

• Conjecture. If L(P) is unbounded as P grows,

then En 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 En (as opposed to show- ing they do not exist.) Understanding the mysterious process whereby the size of Np forces the presence of a rational solutions of En would be a great step forward.

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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-

• ri Perelman, a Russian mathematician.) The
• thers are still open!

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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.

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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.”

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Conclusion Whether it strives for safer internet shopping

• r the elevation of the human spirit, Number

Theory is subject which is rich and fascinating, and replete with many tantalising mysteries.

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