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…
kingdom
the Atlantic,
A long time ago in a galaxy
far,across
far away,
named
a mathematician
Jedi knight named
LukeHardy
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:
1729
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
equals
3
3
1 + 12
equals
3
3
9 + 10
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 a3+b3 and b3+a3 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
YES , sort of , well, actually MAYBE YES, MAYBE NO.
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
3
3
The Taxicab Curve: X + Y = A
“Adding” Points on a Taxicab Curve
X=Y
P
P+Q
Q
R
L
3
3
Taxicab Curve: X + Y = A
“Doubling” a Point on a Taxicab Curve
X=Y
P+P = 2P
P
R
Tangent line to C
at the point P
3
3
Taxicab Curve: X + Y = A
Where is the “Missing” Point?
X=Y
P
P = (a,b) and Q = (b,a)
Q
L
There is no third intersection point!!!
What to do, what to do, what to ................................................?
3
3
Taxicab Curve: X + Y = A
A Pesky Extra Point “at Infinity”
X=Y
Syllogism:
1st
P
Premise: We want a third intersection point.
2nd Premise: Mathematicians always get what they
want.
Q
Conclusion: Since there is no actual third point,
we’ll simply pretend that there is a
L
third point hiding out “at infinity”. We’ll
call that
third
point
O.“at infinity”
O is an
extra
point
It lies on the curve C and the line L.
3
3
Taxicab Curve: X + Y = A
O
A Messy Formula for Adding Points
X=Y
P+Q
P = (x1,y1)
Q = (x2,y2)
R
L
Using a little bit of geometry
and a little bit of algebra, we
can find a formula for the
A( y1  y 2 )  x1x 2 ( y1x 2  y 2 x1 ) A( x1  x 2 )  y1y 2 ( x1y 2  x 2 y1 ) 
Q.

P  Qsum
 of P and
,
 x x (x  x )  y y (y  y )
2
1 2
1
2
 1 2 1
x1x 2 ( x1  x 2 )  y1y 2 ( y1  y 2 ) 
It is a messy formula, but quite practical for computations.
3
3
Taxicab Curve: X + Y = A
An Example: The Taxicab Curve
X3 + Y3 = 1729
Start with Ramanujan’s two points:
P = (1,12) and Q = (9,10)
We can add and subtract them:
 46  37 
PQ  ,

3
3


and
 453  397 
PQ  
,
.
56
56


We can double them and triple them and on and on and on…
 20760  3457 
P  P  2P  
,

1727
1727


and
  5150812031 5177701439 
3P  
,
.
107557668
107557668


 A( y1  y 2 )  x1x 2 ( y1x 2  y 2 x1 ) A( x1  x 2 )  y1y 2 ( x1y 2  x 2 y1 ) 

P  Q  
,
 x1x 2 ( x1  x 2 )  y1y 2 ( y1  y 2 ) x1x 2 ( x1  x 2 )  y1y 2 ( y1  y 2 ) 
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:
identity element
P+O=O+P=P
P + (–P) = O
inverse
P+Q=Q+P
commutative law
(P + Q) + R = P + (Q + R) associative law
Easy to prove
Surprisingly difficult
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 P+Q are also rational
numbers.
This is obvious from the formula for P+Q:
 A( y1  y 2 )  x1x 2 ( y1x 2  y 2 x1 ) A( x1  x 2 )  y1y 2 ( x1y 2  x 2 y1 ) 

P  Q  
,
 x1x 2 ( x1  x 2 )  y1y 2 ( y1  y 2 ) x1x 2 ( x1  x 2 )  y1y 2 ( y1  y 2 ) 
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 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.
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 Points
For example, every rational point on the curve
X3 + Y3 = 7
is equal to some multiple of the single generating point
(2, –1).
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).
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.
Use P to find some points with rational coordinates:
P  2,1
5 4
2P   , 
3 3
3
3
  17 73 
3P  
, 
 38 38 
3
3
 5   4    17   73 
2  (1)        
    7
 3   3   38   38 
3
3
Now multiply everything by 33.383 to clear the denominators!
A Taxicab Curve With Three Integer Points
3
3
3
3
 5   4    17   73 
2  (1)        
    7
 3   3   38   38 
Multiply by 33.383 to clear the denominators!
3
3
(2  3  38)3  ( 3  38)3  (5  38)3  ( 4  38)3
 ( 17  3)3  (73  3)3
 7  33  383
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.
We simply start with four points on the curve X3 + Y3 = 7,
P  2,1
5 4
2P   , 
3 3
  17 73 
3P  
, 
 38 38 
  1256 1265 
4P  
,

 183 183 
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.
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 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.
Taxi(1) = 2
Discovered in:
Taxi(2) = 1729
1657
Taxi(3) = 87539319
1957
Taxi(4) = 6963472309248
1991
Taxi(5) = 48988659276962496
1997
Taxi(6) = 24153319581254312065344
2008
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 we’ve 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 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?
Taxicab Solutions With No Common Factor
Is there a taxicab number A with
two positive no-common-factor solutions?
Yes, Ramanujan gave us one:
1729 = 13 + 123 = 93 + 103.
Is there a taxicab number A with
three 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
equals
5173 + 24683 = 7093 + 24563 = 17333 + 21523
Taxicab Solutions With No Common Factor
How 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 with
five positive no-common-factor solutions?
NO ONE KNOWS!!!!!
A Taxicab Challenge
Find a taxicab number A with five positive nocommon-factor solutions.*
* 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 Bender’s advice:
“Sums of Cubes are everywhere.
Don’t 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