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).
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):
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:
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:
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