Ramanujan Number

The number, 1729, is called the Hardy–Ramanujan number based on a famous anecdote. When GH Hardy visited Ramanujan, he mentioned the number of the cab he rode was 1729. Hardy then remarked that the number didn’t seem special in any way. Wrong, said Ramanujan:
“It is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”
The two different ways are:
1729 = 13 + 123 = 93 + 103
Genius, we think! And yes, Ramanujan was a genius. But can how a genius thinks ever be explained? In this instance, maybe so.

The starting point is this anecdote from Richard Feynman’s book, Surely You're Joking Mr Feynman! Engaged in an arithmetical duel against the abacus, one of the challenges was to come up with cube root of 1729.03. Here’s how Feynman thought:
-         He knew that one cubic foot is 1728 cubic inches, a number very close to 1729.03
-         Therefore, he knew that the cube root of 1728 is 12.
-         And so he started finding the cube root with 12 as the starting point!

What’s the connection to the Ramanjuan number, you wonder.
1)      Well, if Feynman knew of the number 1728 as “one cubic feet”, then chances are very high that so would have many others, including Ramanujan. So it is possible that when Ramanujan heard the number 1729, he (like Feynman) thought of a familiar number close to it: 1728. And realized that 1729 was 13 + 123.
2)     Next, note that the last 3 digits of 1729 are 729 which is 93 (as any mathematician would know). That leaves 1000, which is 103.

Sure, none of us mortals would have reasoned like this. As Stanislas Dehaene said:
“This does not in the least diminish my admiration for Ramanujan’s feats, but it does make him seem more human and understandable.”

Even if this explains how Ramanujan might have thought, as Karl Sabbagh wrote in Dr. Riemann's Zeroes, there’s still one aspect of his comment that can’t be explained:
“Knowing… that 1729 is the smallest number expressible as the sum of two cubes in two different ways. That was quite clever.”

Comments

  1. Ramiah KumarSeptember 8, 2016 at 1:36 PM

    When I was in the US to help our son and daughter-in-law who were expecting a baby, Geetha and I would go the library nearby. I chanced on a book written by an American maths professor for the common person. But this author decided not to hide those equations which sing the glory of discovering mathematicians forever, to cater to the joy of those readers who understand school or college level maths.

    In the context of this blog, I quote from my memory what that professor (mind you a full fledged professional maths person) had to say about our Srinivasa Ramanujan. "It is difficult get over the wonderment of how Ramanujan reeled of amazing equations, much like a magician would pick out live rabbits from his hat!" The examples he had given amazed me. Ramanujan produced an equation which has pi, phi, e - all the three glorious mathematical constants rolled into one equation containing a breath-taking infinite series! No mathematician before him had done that trick of bringing together the 3 constants! No proof was given by Ramanujan but the equation holds! I have now added one more of the equations to my wonderment storage, which I found in the book "The Magic of the Primes" in which Ramanujan produces an equation for a partition number, a high level maths technicality. No mathematician could imagine anything anywhere close to a solution like that till then.

    It is impossible to explain along this blog's lines how Ramanujan went about producing identities that confound mathematicians without giving the proof. Before him only Fermat did that; he drove later mathematicians crazy looking for proof for his identities! When mathematicians started searching for proof for Ramanujan identities, something different happened. While all of Fermat's conjectures were proved right, some of Ramanujan conjectures were proved wrong. But a majority of them sing Ramajunan's glory till date. The research is still going on for some Ramajunan equations. Can you believe Ramanujan got a much sought after maths award more than half a decade after his death, without seeking it naturally!

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