Skewed Ellipses

My diurnal age today is 26793 days and one of the properties of this number is that it's a member of OEIS A118886:

A118886

Numbers expressible as \(x^2 + xy + y^2 \), \(0 \leq x \leq y\), in 2 or more ways.

So in the case of 26793, this means that there are at least two values for \( (x,y) \) such that \(x^2 + xy + y^2 =26793\). It turns out that there are exactly two such values and they are (9, 159) and (93, 96). It's easy to forget that the equation  \(x^2 + xy + y^2 =26793\) is that of an ellipse that is skewed with respect to the \(x\) and \(y\) axes. In other words, it's not your more usual  \(x^2 + y^2 =a^2\) because the \(xy\) term skews it. 

Figure 1 shows the graph along with the two points that have positive integer values: A = (9, 159) and B = (93, 96).

Figure 1

These sorts of ellipses \(x^2 + xy + y^2 =a^2\) have axes of symmetry of \(y=x\) and \(y=-x\) and \( \pm a\) are the intercepts on the \(x\) and \(y\) axes. In the graph shown in Figure 1, the intercepts are \( \pm \sqrt{26793} \approx 163.7 \).

Of course, there are other points on the graph that have integer values but some of these are negative and so are not included in OEIS A118886, Without the condition that \( x \leq y), other points will arise as well. These full range of eight points is shown in Figure 2 and the graph with points added in Figure 3.

Figure 2

Figure 3

Now up to 27000, there are 2043 numbers that qualify for membership in  OEIS A118886 and that represents about 7.6%. However, there are only 69 numbers that are square numbers as well meaning that the intercepts of the \(x\) and \(y\) axes have integer values. These values are:

49, 169, 196, 361, 441, 676, 784, 961, 1225, 1369, 1444, 1521, 1764, 1849, 2401, 2704, 3136, 3249, 3721, 3844, 3969, 4225, 4489, 4900, 5329, 5476, 5776, 5929, 6084, 6241, 7056, 7396, 8281, 8649, 9025, 9409, 9604, 10609, 10816, 11025, 11881, 12321, 12544, 12996, 13689, 14161, 14884, 15376, 15876, 16129, 16641, 16900, 17689, 17956, 19321, 19600, 20449, 21316, 21609, 21904, 22801, 23104, 23716, 24025, 24336, 24649, 24964, 25921, 26569

Take 26569 as an example (see permalink). We have \(26569=159^2\) and the two \( (x,y) \) that satisfy  OEIS A118886 are (0,163) and (75, 112). The full range of ten points are:

(-163, 0), (-163, 163), (-112, -75), (-75, -112), (0, -163), (0, 163), (75, 112), (112, 75), (163, -163) (163, 0)

With these skewed ellipses, if the \(a^2\) in \(x^2 + xy + y^2 =a^2\) is a square number, then there are always another two integer-valued points, \( (a, -a) \) and \( (-a,a) \), compared to non-square numbers. 

Up to 27000, the maximum number of points that satisfy \( 0 \leq x \leq y \) is six. The numbers are 12103, 19747, 22477 and 23569. Using 12103 as an example, the points are (2, 109), (21, 98), (27, 94), (34, 89), (49, 77) and (61, 66). Even extending the range to 100,000, there are no numbers that yield seven points which is not surprising considering the symmetry of the graph. 

There are however, four numbers that yield eight points and these are 53599, 63973, 74347 and 84721. Using, 53599 as an example the eight points are (3, 230), (25, 218), (43, 207), (58, 197), (85, 177), (90, 173), (102, 163) and (122, 145).

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.

Finding Significance in Insignificant Numbers

Sometimes in my examination of the numbers that measure my diurnal age, I come across a number with few entries in the OEIS and all of them inscrutable to my limited intellect. 25197 was a case in point. I wrote about this in an earlier post and what I found in the case of that number was that its square (634888809) contained four consecutive 8's. As such, it belonged to an as yet unidentified sequence of numbers with the common property that their squares contained a sequence of four or more 8's. These numbers are shown below:

This is the sequence that I submitted to the OEIS for approval and I'm still waiting to see it published. This example illustrates that apparently insignificant numbers can contain significance that just needs to be unlocked.

Thus we come to today's number, 25206, that is significant in that it marks a point where there is a balance struck between the number of 4k+1 primes and the number of 4k+3 primes. Prime 617249 marks a point where there are exactly 25206 primes of each sort. While this is clearly significant, the problem is that it's shared with many other nearby numbers as can be seen from this extract from OEIS A092198:

0, 1, 3, 6, 44, 1471, 1472, 1473, 1474, 1475, 1476, 25185, 25187, 25188, 25189, 25190, 25196, 25206, 25211, 25212, 25213, 25214, 25215, 25216, 25217, 25218, 25219, 25222, 25224, 25225, 25251, 25253, 25257, 25258, 25410, 25421, 25426, 25427, ...

Note particularly the gap between 1476 and 25185 (the next member in the sequence). These balance points in the number of 4k+1 and 4k+3 primes clearly exist in clusters. This is the reason that I was looking for something different when examining 25206. It's clear that the number cannot be the sum of two squares because the number factors to 2×3×4201 and there is a 4k+3 prime (3) raised to an odd power (3). However, I wondered if there is an integer solution to the equation \( x^2+2y^2=25206 \) even though there is no integer solution to \( x^2+y^2=25206 \). It turns out there are two, x=158, y=11 and x=38, y=109. This is shown geometrically in the diagram below where the solutions, for positive numbers only, are represented on an ellipse.

The frequency of integer solutions to \( x^2+2y^2=N \) seems about equal to that of  \( x^2+y^2=N \), at least judging by a quick count. For example, over the next twenty integers (25206 to 25216), there are six numbers (25211, 25218, 25219, 25222, 25224, 25225) that can be represented in the form \( x^2+2y^2 \) and six numbers (25209, 25210, 25216, 25220, 25225, 25226) that can be represented in the form  \( x^2+y^2 \). 

From this admittedly quick count, it would seem that about 30% of numbers can be represented as a sum of two squares and another 30% as a sum of a square and twice a square. There is possibly no overlap between the two sets and so it would seem that if a number cannot be represented as a sum of two squares then it may well be possible to represent it as a sum of a square and twice a square. 

This line of enquiry opens up all sorts of intriguing questions such as:

• is it possible to represent a number in both the form \( x^2+y^2 \) and \( x^2+2y^2 \)?
• is it possible to represent all numbers in the form \( ax^2+by^2 \) where \( a \) and \( b \) are positive integers?

2017: A New Year

As the new year begins, it seems appropriate to look at the mathematical character of the number that will identify it: 2017. As a start, this number is prime, in fact the 306th prime. It's nearest neighbours are 2011 and 2027. Thus there will not be another prime year for a decade. Working through the Online Encyclopaedia of Integer Sequences (OEIS), I was made aware initially that the number was linked to one circle and 23 ellipses, each with the major axis equal to the diameter of the circle and major/minor axes coincident with the x and y axes.

The equation all twenty four shapes have in common is \(ax^2+bxy+ cy^2 = 2017\).

In the case of the circle, \(a=1, b=0, c=1\) and so \(x^2 + y^2 = 2017\). The integer solutions are \(x=9\) and \(y=44\). This is the point A on the circle c in the diagram below.

For the ellipses, \(a=1, b=0\) and so \(x^2 + cy^2 = 2017\). The values of \(c\) for which there are integer solutions to \(x\) and \(y\) are 2, 3, 7, 14, 21, 24, 27, 31, 33, 42, 46, 56, 66, 81, 84, 87, 88, 93, 112, 232, 253, 462 and 1848. I've plotted the cases of \(c\)=2, 3, 7 and 14 in the diagram below. The associated points are (37, 18), (17, 24), (15, 16) and (1, 12).

 

For the equation \(ax^2+bxy+ cy^2 = 2017\), when \(a\) is not equal to 1 but \(b\) is still 0, the following \((a, b, c)\) values give integer solutions to \(x\) and \(y\): (4, 0, 9); (2, 0, 41); (5, 0, 17); (2, 0, 65); (2, 0, 95); (8, 0, 65). All these ellipses lie inside the circle \(x^2+y^2=2107\) and have been graphed below (with part of the surrounding circle visible):

If \(xy\) terms are allowed, then there is another whole series of ellipses of the form \(ax^2+bxy+cy^2 = 2017\) where the following values of (a, b, c) yield integer solutions for x and y: (1, 1, 1); (9, 6, 1849); (1, 26, 1); (1, 8, -8); (1, 10, 1); (2, 1, 3); (1, 1, 8); (1, 1, 22); (4, 1, 4); (4, -1, 4); (4, -3, 5); (2, -1, 10); (2, 1, 12); (2, -1, 12); (1, 20, 1); (1, 31, 1); (8, 8, 97); (37, 4, 37); (28, 12, 57); (57, 18, 193). A few of these I've plotted below (with the circle included for comparison):

2017 can also be written as the sum of three (not distinct) cubes, namely \(7^3+7^3+11^3\). Thus it seems that 2107 is unusual in that it can be linked in 2-D space to a large numbers of ellipses. Of course, ellipses are the orbits followed by celestial bodies within the solar system trapped by the gravitational forces of larger bodies such as the Sun and planets.

Consider one of the ellipses above, say \(4x^2+9y^2=2017\) and a general point \((x, y)\) situated on it. The only integer values of \(x\) and \(y\) that satisfy this equation are (±22, ±3) as shown below: