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Counting Lattice Points with GrafEq

By Carlos César de Araújo - cca@gregosetroianos.mat.br 

Portuguese version (with different examples)

We will see on this page how to use GrafEq to count lattice points (i.e., points with integer coordinates) in certain subsets of © Matemática Para Gregos & Troianos - Carlos César. Let us first consider a disk © Matemática Para Gregos & Troianos - Carlos César. It is not dificult to show that the number of lattice points in this region is given by

© Matemática Para Gregos & Troianos - Carlos César

where © Matemática Para Gregos & Troianos - Carlos César is the floor of r. Thus there are

© Matemática Para Gregos & Troianos - Carlos César

pairs © Matemática Para Gregos & Troianos - Carlos César such that © Matemática Para Gregos & Troianos - Carlos César.

We can use GrafEq to visualize this result. All we have to do is to enter with the appropriate set of relations in the algebraic relation window. Equational conditions that characterizes the set © Matemática Para Gregos & Troianos - Carlos César are easy to discover; two of them are © Matemática Para Gregos & Troianos - Carlos César and © Matemática Para Gregos & Troianos - Carlos César. The animation below shows the scanning process used by GrafEq to select the lattice points. You can check that the last frame ends with 221 points. (Press the Escape key to stop the GIF animation.)

Additional constraints can also be included. Below we see the selection of points © Matemática Para Gregos & Troianos - Carlos César in the previous disk for which x and y are relatively prime.

This time we are left with © Matemática Para Gregos & Troianos - Carlos César lattice points. The real answer, nevertheless, is 136. GrafEq missed the points © Matemática Para Gregos & Troianos - Carlos César. What happened? This has to do with the current implementation of gcd in GrafEq 2.11 (the latest version), according to which © Matemática Para Gregos & Troianos - Carlos César is undefined if © Matemática Para Gregos & Troianos - Carlos César. Jeff Tupper will fix this behavior in a future release.

Looking for lattice points in curves is the subject of Diophantine equations. The next animation shows how GrafEq detects the integer solutions to the equation

The graph reveals five lattice points on the hyperbola, namely, © Matemática Para Gregos & Troianos - Carlos César. (Of course, the point © Matemática Para Gregos & Troianos - Carlos César must be excluded.) It is not difficult to prove that these are the only integer solutions.


Carlos César de Araújo,  August 12, 2002

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A giróide (de Alan Shoen) - Animation made with DPGraph and Mathematica by Carlos César

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Mathplotter logo, by Miguel Bayona

instruções

Gráfico de y = mdc(x,1), by Carlos César

Raízes da unidade (made with GrafEq by Carlos César)

A função resto de 1 por x (GrafEq / Carlos César)

Pontos com coordenadas inteiras sob uma hipérbole (Made with GrafEq by Carlos César)

A ciclóide no Winplot (Carlos César)

Parabolóide hiperbólico (veja o applet Java)

Made with Winplot by Carlos César

Made with Winplot by Carlos César