All the Ways of Intuiting 1729

Stanislas Dehaene brings up the Ramanujan-G.H.Hardy anecdote concerning the number 1729. The idea of running through the cubes of all integers from 1 to 12 in order to arrive at Ramanujan's spontaneous recognition of 1729 as the smallest positive integer that can be written in two distinct ways as the sum of two integral cubes is inappropriate and obscures the workings of the naive mathematical mind. To be sure, a computer-mind could come up with that list at a wink. But what would induce it to pop it up when faced with the number 1729 if not prompted by some hunch? Here is a more likely account:

Confronted with 1729 you will recognize at a glance that:

i) 1729 = 1000   (810-81) = 10^3   81 x (10-1)
= 10^3   9^2 x 9
= 10^3   9^3 = (1   9)^3   9^3
= 1   3 x 9   3 x 9^2   9^3   9^3
= 1   (3^3   3 x 3^2 x 9   3 x 3 x 9^2   9^3)
= 1   (3   9)^3
= 1^3   12^3  in view of the pattern
ii) (a   b)^3  
= a^3   3 x a^2 x b   3 x a x b^2                 b^3.    

Now all those 3's in the above expressions spring to attention, you fleetingly call up THE EQUATIONS

iii) (a   b)^3   d^3    
= a^3    (c   d)^3                  a^3   (3 x a^2 x b   3 x a x b^2   b^3)                 d^3                                 
= a^3   (c^3   3 x c^2 x d   3 x c x d^2)                   d^3    

and JUMP to the conclusion that the choice of (1,9; 3,9) for a,b; c,d will give you the smallest positive integer that can be written as the sum the cubes of two integers (a b) and d and also of a different pair a and (c d). You have a well trained instinct. But, if called upon, it will be a simple matter to fill in that jump by a proof, the fixed coefficients 3 ruling out smaller choices for b,c,d, once the minimal possible value 1 is chosen for a.

ANALYSIS OF A TRAIN OF THOUGHT

The best way to understand the process encoded above in technical shorthand is via a metaphor, which should be spun out at leisure. Say you are driving into a strange town, and, for some reason or other, a building complex catches your attention. It does not just pop into your field of vision; at first glance you see it as a museum, a villa, a church or whatever. And then, depending on your particular interests and background, you may recognize its shape, size and purpose, muse over its style, venture a guess as to its vintage, and so forth.

Upon meeting 1729, your first reaction will probably be to break it up into the sum of 1000 and 729, because of our habit of counting in decimal notation. Stop for a moment to consider what would have been facing Ramanujan if Taxi cab companies were favoring binary notation! [11011000001 = 11011000000 1 = 11^3 x 100^3 1^3 = 101^3 x 10^3 11^3 x 11^3 = 1111101000 1011011001]. On the other hand, if you are one of those people obsessed with prime factorization you'll "see" the product 7 x 13 x 19 when somebody says "1729" to you while a before-Thompson-and-Feit but after-Burnside group theorist will say "Aha that is an interesting number, all groups of order 1729 are solvable" and anyone with engineering experience immediately thinks of the 1728 cubic inches contained in a cubic foot [1]. But a historian of Mathematics will see 1729 as the year of Euler's friend and benefactress Catherine the Great's birth.

Next you decide, more or less deliberately, how to investigate the phenomenon. Do you drive to the nearest kiosk, buy a "Baedecker", search for that building and read through all you can find in there about it, before you make up your mind about what you want to know, in other words, assuming you have a kiosk full of lists handy in your own mind, do you run through all the integral cubes smaller than 1729? If so, why cubes?

If you have that kind of mind you probably would first run through the squares before getting to the cubes. The less methodical tourist, eager to enjoy rather than out to complete his (or her) knowledge, may choose to investigate in a haphazard way, spurred on by curiosity, guided by experience, using skills automatically while impulsively following hunches, prowling, sniffing, looking behind bushes, and then jump to rational conclusions.

Now return to Ramanujan and see how the first thing that springs to the naive eye beholding the number 729 is that adding 81 = 9^2 turns it into 810, whereupon 10 drops its disguise, shows one of its true natures as the sum of 1 and 9 and, lo and behold, all those powers of 3 start tumbling in. All the while you are aware of the pattern ii), just below the threshold of consciousness, exactly as a driver is aware of the traffic laws and of the coordinated efforts of his body and his jeep. That is how you find your way through the maze of mathematical possibilities to the "interesting" breakdown of 1729 into two distinct sums of integral cubes.

When you stop to ask yourself what is so great about that, something clicks in your mind: you are facing a positive integer with a certain property, you know that

iv) every collection of positive integers              has a least member
             (in terms of its natural ordering).    

That knowledge, always hovering below the threshold of consciousness, prompts the question whether 1729 might in fact be the LEAST positive integer expressible in distinct ways as the sum of two cubes. Having another look at the representation of 1729 as a sum of various powers of 3 as held in your mind's eye and exhibited in the third line of i) above, the more or less conscious awareness of ii) invites you to break up those sums of cubes according to the pattern iii) where you assume — "without loss of generality" — that a < d = a b, and hence c < b. At this point the solution a = 1, b = c^2 = d and c = 3 surfaces by inspection as "obviously" yielding the minimal value for (a b)^3 d^3.