Taxicabs and

Sums of Two Cubes

An Excursion in Number Theory

Joseph H. Silverman, Brown University

Undergraduate Lecture in Number Theory

Hunter College of CUNY

Tuesday, March 12, 2013

Our Story Begins…

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

a Jedi knight named Luke

received a mysterious package.

kingdom across the Atlantic,

mathematician named Hardy

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.

1729

An Interesting Taxicab Number

1729

equals13 + 123

equals93 + 103

1729 is a sum of two cubes in two different ways

Sums of Two CubesThe 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 a3+b3 and b3+a3 as the same.]

How about four different ways?

And five different ways?

And six different ways?

And seven different ways? …

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.

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 ofYES

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 let’s 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

The Taxicab Curve: X3 + Y3 = A

Taxicab Curve: X3 + Y3 = A

P + Q

“Adding” Points on a Taxicab Curve

P

QR

X = Y

L

Taxicab Curve: X3 + Y3 = A

“Doubling” a Point on a Taxicab Curve

P

Tangent line to C at the point P

R

X = Y

P+P = 2P

Taxicab Curve: X3 + Y3 = A

Where is the “Missing” Point?

X = Y

P

Q

L

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

There is no third intersection point!!!

What to do, what to do, what to ................................................?

Taxicab Curve: X3 + Y3 = A

A Pesky Extra Point “at Infinity”

X = Y

P

Q

Syllogism:

1st Premise: We want a third intersection point.

2nd Premise: Mathematicians always get what theywant.

Conclusion: Since there is no actual third point, we’ll simply pretend that there is a third point hiding out “at infinity”. We’ll call that third point O.

O

L

O is an extra point “at infinity”It lies on the curve C and the line L.

Taxicab Curve: X3 + Y3 = A

P + Q

A Messy Formula for Adding Points

P = (x1,y1)Q = (x2,y2)

R

X = Y

L

Using a little bit of geometry and a little bit of algebra, we can find a formula for the sum of P and Q.

)yy(yy)xx(xx

)yxyx(yy)xx(A,

)yy(yy)xx(xx

)xyxy(xx)yy(AQP

21212121

12212121

21212121

12212121

It is a messy formula, but quite practical for computations.

An Example: The Taxicab CurveX3 + Y3 = 1729

)yy(yy)xx(xx

)yxyx(yy)xx(A,

)yy(yy)xx(xx

)xyxy(xx)yy(AQP

21212121

12212121

21212121

12212121

Start with Ramanujan’s two points:

P = (1,12) and Q = (9,10)

We can add and subtract them:

.56

397,

56

453QP and

3

37,

3

46QP

We can double them and triple them and on and on and on…

.107557668

5177701439,

107557668

5150812031P3 and

1727

3457,

1727

20760P2PP

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

Easy to prove

Surprisingly difficult

In mathematical terminology, the points on the taxicab curve form a GROUP.

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

Curves with an addition law are called

Elliptic Curves

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 P+Q are also rational numbers.

)yy(yy)xx(xx

)yxyx(yy)xx(A,

)yy(yy)xx(xx

)xyxy(xx)yy(AQP

21212121

12212121

21212121

12212121

This is obvious from the formula for P+Q:

The formula is messy, but if A is an integer and if x1, y1, x2, y2 are rational numbers, then the coordinates of P+Q are clearly rational numbers.

The Group of Rational PointsThis 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.

Mordell’s 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 PointsFor 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.

And every rational point on the taxicab curve

X3 + Y3 = 1729

can be obtained by using the two generating points

P = (1,12) and Q = (9,10).

Turning Rational Numbers Into IntegersHow 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.

38

73,

38

17P3

3

4,

3

5P2 1,2P

Use P to find some points with rational coordinates:

738

73

38

17

3

4

3

5)1(2

333333

Now multiply everything by 33.383 to clear the denominators!

A Taxicab Curve With Three Integer Points

738

73

38

17

3

4

3

5)1(2

333333

Multiply by 33.383 to clear the denominators!

33

33

3333

3837

)373()317(

)384()385()383()3832(

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 PointsSuppose that we want a taxicab curve with four integer points.

We simply start with four points on the curve X3 + Y3 = 7,

183

1265,

183

1256P4

38

73,

38

17P3

3

4,

3

5P2 1,2P

and clear their denominators to get a taxicab number

A = 7. 33. 383. 1833 = 63,557,362,007,496

that is a sum of two cubes in four different ways.

And so on. If we start with N points

P, 2P, 3P, 4P, …, NP

and clear all of their denominators, then we will get a (very large) taxicab number that is a sum of two cubes in N different ways.

But Ramanujan Used Positive Integers…That’s 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.

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.

This provides an affirmative answer to our original question.

Taxi(1) = 2Taxi(2) = 1729Taxi(3) = 87539319Taxi(4) = 6963472309248Taxi(5) = 48988659276962496Taxi(6) = 24153319581254312065344

Finding the Smallest Taxicab NumbersThe N’th 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.

Discovered in:16571957199119972008

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…

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 we’ve 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?

it feels as if we’ve cheated.

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.

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 equals

5173 + 24683 = 7093 + 24563 = 17333 + 21523

Is there a taxicab number A withthree positive no-common-factor solutions?

Taxicab Solutions With No Common FactorHow about a taxicab number A with

four positive no-common-factor solutions?

Yes, there’s 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 + 12165003

and 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!!!!!

A Taxicab Challenge

Find a taxicab number A with five positive no-common-factor solutions.

* Or prove that none exist!!!

*

Futurama EpilogueBender is a Bending-Unit:

Chassis # 1729 Serial # 2716057

So take Bender’s advice: “Sums of Cubes are everywhere. Don’t leave home without one!”

Bender's serial number 2716057 is, of course, a sum of two cubes: 2716057 = 952³ + (-951)³.

Taxicabs and

Sums of Two Cubes

Joseph H. Silverman, Brown University

Taxicabs and

Sums of Two Cubes

Joseph H. Silverman, Brown University