Call these (base-10) ellipse numbers. To accurately decide which integers are ellipse numbers, requires an algorithm to decide if n distinct points in Z^2 lie on a unique ellipse. I wouldn't know how to write such an algorithm.
Consider the n-by-6 matrix whose rows are of the form: (1, x_i, y_i, x_i^2, y_i^2, x_i y_i) for i in {1, 2, ..., n}. Now perform Gaussian elimination via row operations. At the end of the process, there should be k linearly-independent rows together with n-k zero rows. The points determine a unique *conic* provided that k = 5 (i.e. the rank of the original matrix is 5). In that case, drop the zero rows to give a 5-by-6 matrix and compute each of the six minors (determinants of 5-by-5 submatrices), labelling them as follows: (A, B, C, D, E, F) Then the equation of the conic is given by: A + B x + C y + D x^2 + E y^2 + F x y = 0 For this to be an ellipse, we require the 'elliptic' constraint: 4 D F > E^2 which is necessary and sufficient for the conic to be an ellipse.
There are also hyperbola numbers, e.g. 13 is a base-10 hyperbola number.
Yes, follow precisely the same algorithm but replace the 'elliptic' constraint with the 'hyperbolic' constraint: 4 D F < E^2 Best wishes, Adam P. Goucher
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Simon Plouffe Sent: Wednesday, June 08, 2016 2:37 PM To: math-fun; Paul Simon; Sylvain Lambert; Louis Plouffe Subject: [math-fun] unusual things
Hello math-funsters,
there is an interesting site here,
http://www.futilitycloset.com/?new=true
a representation of 1/7 in decimal on an ellipse.
the other pages are interesting as well,
quite amusing,
in the same vein ,
that one is original, very original, I counted at least 100000 original images and pages.
http://www.laboiteverte.fr/?s=math
like this 'etch-a-sketch' on a sphere :
http://www.laboiteverte.fr/le-doodle-dome-de-tyco-lardoise-magique- spherique/
and with Pi digits here ??:
http://www.laboiteverte.fr/le-doodle-dome-de-tyco-lardoise-magique- spherique/#jp-carousel-68129
This is what I call original.
one drawback : the whole site is in french, not the images.
Have fun.
Cheers.
Simon Plouffe
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