Was Ramanujan wrong, or wrongly reported?

 Most recreational mathematicians know the story of Godfrey Hardy’s taxi. In brief, Hardy called on his sick colleague, Srinivasa Ramanujan. In the course of making conversation, Hardy mentioned the number of his taxi-cab, his favourite form of transport. It had, said Hardy, a rather dull number, 1729. “No, Hardy! No, Hardy!” replied Ramanujan, “It is a very interesting number - it is the smallest number expressible as the sum of two cubes in two different ways.”

Ramanujan was referring here to the fact that 1729 is the sum of one cubed and twelve cubed, and also the sum of nine cubed and ten cubed. The two mathematicians then went on to discuss the fourth powers equivalent, but that has no part here. There is a solution, by the way, with 133 and 134 being the numbers on one side: the rest I leave to you, once you have my methodology, set out below. So Hardy is mainly remembered by mathematicians as the person who played straight man to Ramanujan.

There was more, as we shall see, but first, a small diversion: 1729 is one of a special group of numbers called Carmichael numbers, which are important in number theory. It is highly likely that Hardy was trying to find out if Ramanujan had discovered these numbers in his intuitive way, and got an answer from left field instead. As I am about to reveal, though, this was wrong, and given Ramanujan’s brilliance, it is far more likely that he was misquoted

It has been known for thirty years or so that there is an infinite number of Carmichael numbers, but is there an infinite number of them with factors in arithmetic progression? That description fits 1729 (7 x 13 x 19), but that may be just happenstance. On the other hand, I read recently that 91 is expressible as the sum of two cubes in two different ways: 91 = 3³ + 4³ = (-5)³ + 6³. 

At 0600 this morning, it was dark, I had fetched the newspaper, and was trying to remember the target number, and the cubes that composed it. Then I recalled that it was 91, which my mind had filed as interesting, because it is ¹/₁₉ of 1729, being 7x13.

That did it. I got up, fired up Excel, and set to work. But before I continue, what are Carmichael numbers? Mathematicians will understand when I note that there is insufficient space in the margin of the page to offer it in full…

OK, I won’t be mean to those interested but less familiar with the trivia. Pierre de Fermat (1601–1665) is  remembered today mainly for his “Last Theorem”, which took more than 300 years to prove. In the margin of his copy of Diophantus’ Arithmetica, Fermat wrote:

“To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.”

Now on with the spreadsheet and how I saved myself a lot of what we Australians call hard yakka. One way to solve knotty problems is to try all the possibilities and these are Diophantine solutions, named after the author of the book that Fermat scribbled his note in.

I once wrote in one of my books that Diophantus would have killed to get his hands on a computer and a spreadsheet program, and I meant it. I am still trying to find a way of using a spreadsheet to test Collatz' conjecture.

In cell A2, I entered the value -20, then I selected that column down to row 89, and used FILL – SERIES to integers down to 67. Next, in cell B2, I inserted this formula: =A2*A2*A2. This, of course, returns the value (-8000), being the cube of -20.

Next, I used COPY – DOWN, or CTRL-D, to fill column B with cubes. Then I was ready to laboriously typed in the first row: C2 (=B2+B3); D2 (=B2+B4); E2 (=B2+B5) and so on, all the way to column AQ. Then I could highlight rows 2 to 89 and columns C to AQ and fill those cells with COPY – DOWN, or CTRL-D.

As you can see, I now had more sums-of-two-cubes values than I could poke a stick at, and a few of my “hits” are marked with colour. I highlighted all of the values, copied them and did an unformatted paste into a Word file. This gave me tab delimited rows, so I had to get rid of the tabs. In Word, CTRL-h gives FIND AND REPLACE, and if you are smart-lazy like me, you either know, or need to know two codes to use. A tab marker is ^t, and a carriage return (end of paragraph) is ^p.

So in no time at all, I had 1845 values that could be sorted into numerical order and searched. After getting through less than a page, I muttered something that a passing kookaburra misheard as beggar this for a game of soldiers. The actual words are now lost to the mists of time, so we shall move on.

I highlighted the whole column (CTRL-A) and copied it (CTRL-C). Then back to the spreadsheet, open a new worksheet, click on A1 and paste (CTRL-V). Now I have all of my values in order, but no great desire to eyeball them, as I had had no breakfast, and no mug of tea, either. Time for smart-lazy again. In cell B2, I added this formula: =IF(A1=A2, "hit","").

 

As you can see, there was no need to scrutinise all the values, but look on the right, where there are some trebles. Now ignoring zero, which can be obtained in an infinite number of ways: x³ + (-x)³, where x is any integer, the first treble is well below Ramanujan’s 1729. You can get both 728 and -728 in three ways.

Here they are: 728 = (-10)³ + 12³ = 6³ + 8³ = (-1)³ + 9³

Numerology is a trap, a snare and a delusion, but the difference between 1729 and 728 is 1001 (7x11x13), while 91 is 7x13 and 1729 is, as noted above, 7, 13 and 19. You can see why people get drawn in, even if I don’t mention that only in base-13 notation is it true that 6x9=42!

He proves by algebra that Hamlet's grandson is Shakespeare's grandfather and that he himself is the ghost of his own father.
—James Joyce, Ulysses, 21.

I think I’ll stop there.