The “beauty” Of Number Theories

By Author: Khalil A. Cassimally
Total Articles: 8
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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.