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(After-math of last week’s second year lecture on elliptic
curves.) We all know the story of
Ramanujan and the
taxicab, immortalized by
Hardy

I
remember once going to see him when he was lying ill at Putney. I had
ridden in taxicab no. 1729 and remarked that the number seemed to me
rather a dull one, and that I hoped it was not an unfavorable omen.
‘No,’ he replied, ‘it’s a very interesting number; it is the smallest
number expressible as a sum of two cubes in two different ways’.

When I was ten, I wanted to become an archeologist and
even today I can get pretty worked-up about historical facts. So, when I
was re-telling this story last week I just had to find out things like :
the type of taxicab and how numbers were displayed on them and, related
to this, exactly when and where did this happen, etc. etc. Half an hour
free-surfing further I know a bit more than I wanted. Let’s start with
the date of this taxicab-ride, even the year changes from source to
source, from 1917 in the dullness of 1729 (arguing that Hardy
could never have made this claim as 1729 is among other things the third
Carmichael Number, i.e., a pseudoprime relative to EVERY base) to ‘late
in WW-1′ here… Between 1917
and his return to India on march 13th 1919, Ramanujan was in and out a
number of hospitals and nursing homes. Here’s an attempt to summarize
these dates&places (based on the excellent paper Ramanujan’s Illness by D.A.B. Young). (may 1917 -
september 20th 1917) : Nursing Hostel, Thompson’s Lane in Cambridge.
(first 2 a 3 weeks of october 1917) : Mendip Hills Senatorium, near
Wells in Somerset. (november 1917) : Matlock House Senatorium at
Matlock in Derbyshire. (june 1918 – november 1918) : Fitzroy House, a
hospital in Fitzroy square in central London. (december 1918 – march
1919) : Colinette House, a private nursing home in Putney, south-west
London. So, “he was lying ill at Putney” must have meant that Ramanujan
was at Colinette House which was located 2, Colinette Road and a quick
look with Google Earth

shows that the
The British Society for the History of Mathematics Gazetteer
is correct in asserting that “The house is no longer used as a nursing
home and its name has vanished” as well as

It was in 1919
(possibly January), when Hardy made the famous visit in the taxicab
numbered 1729.

Hence, we are looking for a London-cab
early 1919. Fortunately, the London Vintage Taxi
Association has a website including a
taxi history page.

At
the outbreak of the First World War there was just one make available to
buy, the Unic. The First World War devastated the taxi trade.
Production of the Unic ceased for the duration as the company turned to
producing munitions. The majority of younger cabmen were called up to
fight and those that remained had to drive worn-out cabs. By 1918 these
remnant vehicles were sold at highly inflated prices, often beyond the
pockets of the returning servicemen, and the trade deteriorated.

and as the first post-war taxicab type was introduced in
1919 (which became known as the ‘Rolls-Royce of cabs’) more than likely
the taxicab Hardy took was a Unic

and the number
1729 was not a taxicab-number but part of its license plate. I still
dont know whether there actually was a 1729-taxicab around at the time,
but let us return to mathematics. Clearly, my purpose to re-tell the
story in class was to illustrate the use of addition on an elliptic
curve as a mean to construct more rational solutions to the equation
$x^3+y^3 = 1729 $ starting from the Ramanujan-points (the two
solutions he was referring to) : P=(1,12) and Q=(9,10). Because the
symmetry between x and y, the (real part of) curve looks like

and if we take
0 to be the point at infinity corresponding to the asymptotic line, the
negative of a point is just reflexion along the main diagonal. The
geometric picture of addition of points on the curve is then summarized
in

and sure

enough we found the points

$P+Q=(\frac{453}{26},-\frac{397}{26}) $ and

$(\frac{2472830}{187953},-\frac{1538423}{187953}) $ and so on

by hand, but afterwards I had the nagging feeling that a lot more could

have been said about this example. Oh, if Im allowed another historical

side remark : I learned of this example from the excellent book by Alf

Van der Poorten Notes on Fermat’s last theorem page 56-57. Alf acknowledges that he

borrowed this material from a lecture by Frits Beukers ‘Oefeningen rond

Fermat’ at the National Fermat Day in Utrecht, November 6th 1993.

Perhaps a more accurate reference might be the paper Taxicabs and sums of two cubes by Joseph

Silverman which appeared in the april 1993 issue of The American

Mathematical Monthly. The above drawings and some material to follow is

taken from that paper (which I didnt know last week). I could have

proved that the Ramanujan points (and their reflexions) are the ONLY

integer points on $x^3+y^3=1729 $. In fact, Silverman gives a

nice argument that there can only be finitely many integer points on any

curve $x^3+y^3=A $ with $A \in \mathbb{Z} $ using the

decomposition $x^3+y^3=(x+y)(x^2-xy+y^2) $. So, take any

factorization A=B.C and let $B=x+y $ and

$C=x^2-xy+y^2 $, then substituting $y=B-x $ in the

second one obtains that x must be an integer solution to the equation

$3x^2-3Bx+(B^2-C)=0 $ Hence, any of the finite number of

factorizations of A gives at most two x-values (each giving one

y-value). Checking this for A=1729=7.13.19 one observes that the only

possibilities giving a square discriminant of the quadratic equation are

those where $B=13, C=133 $ and $B=19, C=91 $ leading

exactly to the Ramanujan points and their reflexions! Sure, I mentioned

in class the Mordell-Weil theorem stating that the group of rational

solutions of an elliptic curve is always finitely generated, but wouldnt

it be fun to determine the actual group in this example? Surely, someone

must have worked this out. Indeed, I did find a posting to

sci.math.numberthy by Robert L. Ward : (in fact, there is a nice page on

elliptic curves made

from clippings to this newsgroup) From: rlward1@orion.ncsc.mil

(Robert L. Ward)
Subject: Re: the MW group of the taxicab cubic

Date: 10 Nov 99 16:16:22 GMT
Newsgroups:

sci.math.numberthy
I used MAGMA to answer this question. The

minimal model of this curve is: Y^2 + Y = X^3 - 20178727, its

discriminant is -3^9_7^4_13^4*19^4, and its conductor is 80714907 =

3^3_7^2_13^2*19^2. The transformation to get this equation from x^3 +

y^3 = 1729 is X = 5187/(y+x),
Y = (7780_y-7781_x)/(x+y),<br

/> x = (7780-Y)/(3_X),
y = (7781+Y)/(3_X).
I issued

the following commands: E := EllipticCurve([0,1,0,0,-20178727]);<br

/> MW,h := MordellWeilGroup(E);
P := h(MW.1);
Q :=

h(MW.2);
Then P turned out to be (273,409) and Q to be (399,6583),

which correspond to the Ramanujan solutions of the original equations.

Thus MAGMA asserts that generators of the Mordell-Weil group of the

curve are indeed the Ramanujan points. Robert L. Ward This

would be great! all rational solutions of the taxicab-curve are

constructed from the Ramanujan points by addition on the elliptic

curve. However, the lost archeologist in me wanted to check

this himself… Ive never done calculations with elliptic curves in

sage so now was a good time

to learn this. Sage assumes to input your elliptic curve in

Weirstrass-form but fortunately Roberts transformations above transfer

the taxicab-curve to Weierstrass form $y^2+y=x^3-20178727 $

and sage accepts the array-input $(a_1,a_2,a_3,a_4,a_6) $ of an

elliptic curve defined by $y^2+a_1xy+a_3y = x^3+a_4x+a_6 $ and

sage has some very powerful routines for elliptic curves, including John

Cremona’s

Mwrank

function to compute the rank of the Mordell-Weil group and it can also

determine the torsion part and generators of the Mordell-Weil group. So,

here we go (and it returned it all in some 5 minutes on my MacBook)

sage: T=EllipticCurve([0,0,1,0,-20178727])
sage:

T.torsion_subgroup()
Trivial Abelian Group
sage: T.rank()

2
sage: T.gens()
[(273 : 409 : 1), (1729 : 71753

1)] The Mordell-Weil group of the taxicab-curve is isomorphic
to $\mathbb{Z} \oplus \mathbb{Z} $ and the only difference with
Robert Wards posting was that I found besides his generator
$P=(273,409) $ (corresponding to the Ramanujan point (9,10)) as
a second generator the point $Q=(1729,71753) $ (note again the
appearance of 1729…) corresponding to the rational solution $(
-\frac{37}{3},\frac{46}{3}) $ on the taxicab-curve. Clearly, there
are several sets of generators (in fact that’s what
$GL_2(\mathbb{Z}) $ is all about) and as our first generators
were the same all I needed to see was that the point corresponding to
the second Ramanujan point (399,6583) was of the form $\pm Q + a
P $ for some integer a. Points and their addition is also easy to do
with sage sage: P=T([273,409])
sage: Q=T([1729,71753])

sage: -P-Q
(399 : 6583 : 1) and we see that the
second Ramanujan point is indeed of the required form!