In the proceedings of the London Mathematical Society of May
1921, Srinivasa Ramanujan said something startling as a reply to
G.H. Hardy?s suggestion that the number of a taxi-cab (1729) was
?dull.? Ramanujan?s reply was as follows: ?No, it is a very
interesting number; it is the smallest number expressible as a
sum of two cubes in two different ways.?
?Wow,? I remember thinking when I first read that. And he was
right of course. Ramanujan spotted the two sets of cubes being
13 +123 and 93+103 respectively. Grab the nearest calculator and
check the results of these two expressions. Scary?
Ramanujan is today considered as the father of modern number
theory by many mathematicians. He made a number of original
discoveries in number theory, especially in collaboration with
G.H Hardy, a theorem concerning the partition of numbers into a
sum of smaller integers. For example the number 4, has five
partitions as it can be expressed in five ways which are ?4,?
?3+1,? ?2+2,? ?2+1+1? and ?1+1+1+1.?
Ramanujan made partition lists for the first 200 integers in his
tattered notebook and observed a strange regularity. For any
number that ends with the digit 4 and 9, the number of possible
partitions is always divisible by 5. Secondly, starting with the
number 5, the number of partitions for every seventh integer is
a multiple of 7. And thirdly, starting with 6, the partitions
for every eleventh integer are a multiple of 11.
These strange numerical relationships that Ramanujan discovered
are now called the three Ramanujan ?congruences.? And these
relationships completely shocked the mathematical community: the
multiplicative behaviours should apparently have had nothing to
do with the additive structures involved in partitions.
However during the Second World War, Freeman Dyson, a
mathematician and physicist, developed a tool that allowed him
to break partitions of whole numbers into numerical groups of
equal sizes. According to Dyson, this tool, which he called
?rank,? would be able to prove the three Ramanujan congruences.
Unluckily for Dyson though, rank worked only with 5 and 7 but
not with 11. He had however made a huge step in proving
Ramanujan?s findings. But the problem now seemed to be with 11,
right? Partly!
In the 1980s, the mystery of 11 was finally solved by 2 other
mathematicians, Andrews and Garvan. But the story did not end
there.
In the late 1990s, completely by chance, Ken Ono, and expert on
Ramanujan?s work, came upon one of Ramanujan?s original tattered
notebooks. In there he notice a peculiar numerical formula that
seemed to have no link whatsoever with partitions. The formula
however proved to be the spinner.
Working with the formula, Ono proved later that partition
congruences do not only exist for 5,7 and 11 but could also be
found for all larger primes. So now would Andrews?s and Garvan?s
tool (called ?crank?), inspired from Dyson?s ?rank,? work with
all those infinite number of partition congruences?
Well yes. Karl Mahlburg, a young mathematician has spent a whole
year manipulating numerical formulae and functions that came out
when he applied ?crank? on various prime numbers. Mahlburg says
he slowly started spotting uniformity between the formulae and
functions. And then came the brainstorm or ?fantastically clever
argument,? as Ono puts it.
Basing his owns work on Ono?s, Mahlburg discovered that the
partition congruence theorem still holds if the partitions were
broken down in a different manner. Instead of breaking the
number 115 for example into five equal partitions of 23 (which
is not a multiple of 5), he split the number into 25, 25, 25, 30
and 10. As each part is a multiple of 5, it follows that the sum
of the parts is also a multiple of 5. In fact Mahlburg showed
that this concept extends to every prime number thereby proving
that Andrews?s and Garvan?s crank worked for all those infinite
number of partition congruences.
Incredible how such number theories, which are so full of wit,
may be as difficult as this to prove. It did take about a
complete century to prove Ramanujan?s partition congruences
anyway.
But coming back to the taxi-cab number 1729. This number is
nowhere near ?dull.? (9-7)/2=1 9-(2/1)=7 (9/1)-7=2 (7/1)+2=9
Cool!
About the author:
Khalil A. Cassimally is currently Senior Columnist at
BackWash.com and Columnist for bbc.co.uk h2g2 The Post where he
writes 'Not Scientific Science' column.
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