ramanujan 1729 taxicab

1. 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

2. 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.

3. 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.

4. 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.

5. An Interesting Taxicab Number
1729
equals
13
+ 123
equals
93
+ 103
1729
is a sum of two cubes in two different ways

6. 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.]
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
.

7. 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

8. 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.

9. 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?

10. The Taxicab Curve: X3
+ Y3
= A
The Taxicab Curve: X3
+ Y3
= A

11. Taxi(1) = 2
Taxi(2) = 1729
Taxi(3) = 87539319
Taxi(4) = 6963472309248
Taxi(5) = 48988659276962496
Taxi(6) = 24153319581254312065344
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.
Discovered in:
1657
1957
1991
1997
2008
Maybe you can find the next one!

12. 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.

13. 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
.
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 with
three positive no-common-factor solutions?

14. 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

15. 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 no-
common-factor solutions.
* Or prove that none exist!!!
*

16. Futurama Epilogue
Bender 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.

17. Taxicabs
and
Sums of Two Cubes
Joseph H. Silverman, Brown University