The Original Taxi Cab Number in a New Light

Today I turned 25829 days old and, amongst the number's many different properties, one in particular caught my eye. The property was that it is a member of OEIS A262054: Euler pseudoprimes to base 7: composite integers such that:$$ |7^{(n-1)/2}| \equiv 1 \pmod {n}$$Now there's no sign of 1729, the original taxi cab number, but we'll get there. Firstly however, how did 1729 earn its sobriquet? Here an excerpt from Wikipedia:

The name is derived from a conversation in about 1919 involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy: 

"I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways."

Figure 1

The two different ways are: \(1^3 + 12^3\) and \(9^3 + 10^3\).

I won't go further into taxicab numbers here as the Wikipedia article explains things well enough. What I want to do is cast a new light on 1729, the number that Hardy originally thought was a rather dull number. The light I'm casting comes from the Euler pseudoprimes. 

I've already discussed pseudoprimes in two earlier posts: Fermat Pseudoprimes and Carmichael Numbers. I did make make passing mention of Euler pseudoprimes in the former post but didn't go into the matter further. Figure 1 shows a screenshot of part of what Wikipedia has to say about Euler pseudoprimes. 

The excerpt in Figure 1 concludes with the observation that:

The absolute Euler pseudoprimes are a subset of the absolute Fermat pseudoprimes, or Carmichael numbers, and the smallest absolute Euler pseudoprime is 1729 = 7×13×19.

It's surprising then that a mathematician of Hardy's calibre should not have recognised 1729 as having quite some claim to fame. In fact, the Online Encyclopaedia of Integer Sequences (OEIS) has 794 entries for the number so it is far from dull. Let's consider some of the other entries for 1729 in the OEIS. 

One entry is not surprising when the factorisation of 1729 is considered and the factors are arranged in descending order: 19 x 13 x 7. Let's add a 1 to give 19 x 13 x 7 x 1. The numbers 19, 13, 7 and 1 form an arithmetic sequence and this shows 1729 to be a so-called sextuple factorial. This can be written as 19!!!!!! or 19!6.

1729 counts the ways that a 2 x 2 matrix can be populated with integers from -7 to +7 in such a way that every matrix is singular (that is has a determinant of zero). It thus forms part of OEIS A209981. 

Figure 2: 35 points in a body-centered cubic lattice, 
forming two cubical layers around a central point

In the realm of figurate numbers, 1729 is a centred cube number. These are numbers of the form:\((n+1)^3+n^3\) and of course 1729 can be written as \(10^3+9^3\). Figure 2 shows the example of the centred cube number 35 and Wikipedia explains:

A centred cube number is a centred figurate number that counts the number of points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with \(i^2\) points on the square faces of the \(i\)-th layer. Equivalently, it is the number of points in a body-centred cubic pattern within a cube that has \(n + 1\) points along each of its edges. 

The first few centred cube numbers are:

1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, ... (sequence A005898 in the OEIS).

1729 is also an heptagonal number, a 12-gonal or dodecagonal number, a 24-gonal or icosotetragonal number but that's probably enough for the moment.

Mathematics in Everyday Life

How many times have I opened a box of tissues by removing the elliptical cover on the top of the box? Every time I do it, I'm aware of its elliptical shape but I never paused to consider the resulting ellipse of cardboard that I held in my hand. It would be discarded as rubbish. Today however, I paused and really looked at what I had in my hand (see Figure 1).

Figure 1

There's even a little semi-circular tab on the right that can be depressed to facilitate the removal of the cover. I'd never noticed that before. Turning the cardboard ellipse over reveals blank cardboard on which I marked in the major and minor axes and measured their lengths, to the nearest millimetre (see Figure 2).

Figure 2

These measurements enable calculation of the eccentricity \(e\) of the ellipse and so in this case, with \(a=28\) and \(b=62.5\) where \(a\) and \(b\) are the lengths of the semi-minor and semi-major axes respectively, we have:$$e=\sqrt {1-\frac{a^2}{b^2}}=\sqrt {1-\frac{28^2}{62.5^2}} \approx 0.894$$This of course is highly elliptical, especially if it's compared with the eccentricities of the planets of the solar system (see Figure 3).

Figure 3

As can be seen in Figure 3, Mercury and Pluto have the most eccentric orbits but much less eccentric than my cardboard ellipse. The other planets have elliptical orbits that would be hard to distinguish from circles if their proportions were displayed on a cardboard cut-out similar to that shown in Figure 2. Coincidentally, there is a centaur with an eccentricity of 0.894 as the table shown in Figure 4 reveals. The academic paper that the table was taken from is quite an interesting but I won't go into here but this is the link, the same as the one shown in Figure 4.

Figure 4

As explained in Figure 4, centaurs are planetesimals with perihelia (closest distance to the Sun) exterior to the orbit of Jupiter and aphelia (farthest distance from the Sun) interior to the orbit of Neptune. The most famous of the centaurs in Chiron, the first to be discovered in 1977 but the somewhat less famous C/2012 H2 (McNaught) does have an orbit that exactly matches that of the cardboard ellipse shown in Figures 1 and 2. Figure 5 provides a little more information about this object.

Figure 5

To calculate the length \(F\) from the centre of the ellipse to the two foci, the following formula can be used involving once again the lengths of the semi-minor and semi-major axes:$$F=\sqrt{b^2-a^2}=\sqrt{62.5^2-28^2} \approx 55.9$$These foci for the cardboard ellipse are shown in Figure 6.

Figure 6

The mathematics in this post is very basic but that was my intention. Though basic, the shape of the cardboard ellipse is nonetheless reflected in the shape of a particular centaur's orbit and it's pretty cool to find a connection between an everyday household item and the solar system in which we are immersed.