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Taxicabs and Sums of Two Cubes

- An Excursion in Number Theory

Undergraduate Lecture in Number Theory Hunter

College of CUNY Tuesday, March 12, 2013

Joseph H. Silverman, Brown University

Our Story Begins

A long time ago in a galaxy far, far away, a

Jedi knight named Luke received a mysterious

package.

This package, from a young Indian office clerk

named

Ramanujan,contained pages of scribbled

mathematical formulas. Some of the formulas were

well-know exercises. Others looked preposterous

or wildly implausible. But Hardy and a colleague

managed to prove some of these amazing formulas

and they realized that Ramanujan was a

mathematical genius of the first order.

Our Story Continues

Hardy arranged for Ramanujan to come to

England. Ramanujan arrived in 1914 and over the

next six years he produced a corpus of brilliant

mathematical work in number theory,

combinatorics, and other areas. In 1918, at the

age of 30, he was elected a Fellow of the Royal

Society, one of the youngest to ever be

elected. Unfortunately, in the cold, damp climate

of England, Ramanujan contracted tuberculosis.

He returned to India in 1920 and died shortly

thereafter.

A Dull Taxicab Number

Throughout his life, Ramanujan considered numbers

to be his personal friends. One day when

Ramanujan was in the hospital, Hardy arrived for

a visit and remarked The number of my taxicab

was 1729. It seemed to me rather a dull

number. To which Ramanujan replied No, Hardy!

It is a very interesting number. It is the

smallest number expressible as the sum of two

cubes in two different ways.

An Interesting Taxicab Number

1729

1729 is a sum of two cubes in two different ways

Sums of Two Cubes

The taxicab number 1729 is a sum of two cubes in

two different ways. Can we find a number that is

a sum of two cubes in three different ways? When

counting solutions, we treat a3b3 and b3a3 as

the same.

The answer is yes 4104 163 23 153 93

(12)3 183.

Of course, Ramanujan really meant us to use only

positive integers 87,539,319 4363 1673

4233 2283 4143 2553.

How about four different ways? And five

different ways? And six different ways?

And seven different ways?

Sums of Two Cubes in Lots of Ways

Motivating Question Are there numbers that can

be written as a sum of two (positive) cubes in

lots of different ways?

The answer, as we shall see, involves a

fascinating blend of geometry, algebra and number

theory.

And at the risk of prematurely revealing the

punchline, the answer to our question is

, well, actually MAYBE YES, MAYBE NO.

, sort of

YES

Taxicab Equations and Taxicab Curves

Motivating Question as an Equation Are there

numbers A so that the taxicab equation X3 Y3

A has lots of solutions (x,y) using (positive)

integers x and y?

Switching from algebra to geometry, the

equation X3 Y3 A describes a taxicab curve

in the XY-plane.

The Geometry of a Taxicab Curve

So lets start with an easier question. What are

the solutions to the equation X3 Y3 A in real

numbers?

In other words, what does the graph of X3 Y3

A look like?

The Taxicab Curve X3 Y3 A

Adding Points on a Taxicab Curve

P Q

Doubling a Point on a Taxicab Curve

PP 2P

Tangent line to C at the point P

Where is the Missing Point?

P (a,b) and Q (b,a)

L

There is no third intersection point!!!

What to do, what to do, what to

................................................?

A Pesky Extra Point at Infinity

Syllogism 1st Premise We want a third

intersection point. 2nd Premise Mathematicians

always get what they want. Conclusion Since

there is no actual third point, well simply

pretend that there is a third point hiding out

at infinity. Well call that third point O.

L

O is an extra point at infinityIt lies on the

curve C and the line L.

A Messy Formula for Adding Points

P Q

Using a little bit of geometry and a little bit

of algebra, we can find a formula for the sum of

P and Q.

It is a messy formula, but quite practical for

computations.

An Example The Taxicab CurveX3 Y3 1729

Start with Ramanujans two points P (1,12)

and Q (9,10)

Is Addition Really Addition?

Adding points on the taxicab curves is certainly

very different from ordinary addition of numbers.

But taxicab addition and ordinary addition do

share many properties.

Let O denote the extra point at infinity and

for any point P (x,y) on the curve, let P be

the reflected point (y,x).

Properties of Taxicab Addition P O O P

P P (P) O P Q Q P (P Q) R P (Q

R)

identity element inverse commutative

law associative law

In mathematical terminology, the points on the

taxicab curve form a GROUP.

Elliptic Curves

Curves with an addition law are called Elliptic

Curves

- Elliptic curves and functions on elliptic curves

play an important role in many branches of

mathematics and other sciences, including - Number Theory
- Algebraic Geometry
- Cryptography
- Topology
- Physics

Adding Rational Points Gives More Rational

Points

Taxicab addition has one other very important

property

If the coordinates of P (x1,y1) and Q

(x2,y2) are rational numbers, then the

coordinates of PQ are also rational numbers.

The formula is messy, but if A is an integer and

if x1, y1, x2, y2 are rational numbers, then the

coordinates of PQ are clearly rational numbers.

The Group of Rational Points

This means that we can add and subtract points in

the set C(Q) (x,y) ? C x and y are

rational numbers ? O and stay within this

set. Thus C(Q) is also a group.

One of the fundamental theorems of the 20th

century says that we can get every point in C(Q)

by repeated addition and subtraction using a

finite starting set.

Mordells Theorem (1922) There is a finite set

of points P1, P2, , Pr in C(Q) so that

every point in C(Q) can be found by repeatedly

adding and subtracting P1, P2, , Pr. In other

words, for every point P in C(Q), we can find

integers n1,n2,, nr so that P n1P1 n2P2

nrPr.

Examples of Groups of Rational Points

For example, every rational point on the curve X3

Y3 7 is equal to some multiple of the single

generating point (2, 1).

So we now understand how to find lots of

solutions to X3 Y3 A using rational numbers x

and y, but our original problem was to find lots

of solutions using integers.

Turning Rational Numbers Into Integers

How can we change rational numbers into integers?

Answer Multiply by a common denominator.

Start with the point P (2, 1) on the curve X3

Y3 7.

Now multiply everything by 33.383 to clear the

denominators!

A Taxicab Curve With Three Integer Points

Multiply by 33.383 to clear the denominators!

We have constructed a taxicab number A 7.33.383

10,370,808 that is a sum of two cubes in three

different ways

2283 (114)3 1903 1523 ( 51)3 2193

10,370,808

Taxicab Curves With Lots of Integer Points

Suppose that we want a taxicab curve with four

integer points.

But Ramanujan Used Positive Integers

Thats okay, because it is possible to prove that

in the list of points P, 2P, 3P, 4P, 5P, 6P,

7P, there are infinitely many of them whose x

and y coordinates are both positive.

So we can take N of these positive points from

the list and clear all their denominators.

This provides an affirmative answer to our

original question.

Pick any number N. Then we can find a taxicab

number A so that the taxicab equation X3 Y3

A has at least N different solutions (x,y) using

positive integers x and y.

Finding the Smallest Taxicab Numbers

The Nth Taxicab Number is the smallest number A

so that we can write A as a sum of two positive

cubes in at least N different ways.

It is not easy to determine Taxi(N) because the

numbers get very large, so it is hard to check

that there are no smaller ones.

Here is the current list.

Taxi(1) 2Taxi(2) 1729Taxi(3)

87539319Taxi(4) 6963472309248Taxi(5)

48988659276962496Taxi(6) 2415331958125431206534

4

Discovered in16571957199119972008

Maybe you can find the next one!

Are We Really Done?

What we have done is take a lot of solutions

using rational numbers and cleared their

denominators. This answers the original question,

but

it feels as if weve cheated.

Suppose that we want to find taxicab numbers that

are truly integral and that do not come from

clearing denominators. How can we tell if weve

cheated? Well, if A comes from clearing

denominators, then the x and y values will have a

large common factor.

New Version of the Motivating Question Are there

taxicab numbers A for which the equation X3

Y3 A has lots of solutions (x,y) using positive

integers so that x and y have no common factor?

Taxicab Solutions With No Common Factor

Is there a taxicab number A withtwo positive

no-common-factor solutions?

Yes, Ramanujan gave us one 1729 13 123 93

103.

Is there a taxicab number A withthree positive

no-common-factor solutions?

Yes, Paul Vojta found one in 1983. At the time he

was a graduate student and he discovered this

taxicab number using an early desktop IBM PC!

15,170,835,645 equals5173 24683 7093

24563 17333 21523

Taxicab Solutions With No Common Factor

How about a taxicab number A withfour positive

no-common-factor solutions?

Yes, theres one of those, too, discovered

(independently) by Stuart Gascoigne and Duncan

Moore just 10 years ago.

1,801,049,058,342,701,083 equals 922273

12165003and 1366353 12161023 and 3419953

12076023 and 6002593 11658843

Taxicab Solutions With No Common Factor

Is there a taxicab number A withfive positive

no-common-factor solutions?

NO ONE KNOWS!!!!!

Or prove that none exist!!!

Futurama Epilogue

- Bender is a Bending-Unit Chassis 1729 Serial

2716057

Bender's serial number 2716057 is, of course, a

sum of two cubes 2716057 952³ (-951)³.

So take Benders advice Sums of Cubes are

everywhere. Dont leave home without one!

Taxicabs and Sums of Two Cubes

Joseph H. Silverman, Brown University

Taxicabs and Sums of Two Cubes

Joseph H. Silverman, Brown University

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