Lattice Points on Lines

The point of this project is to find out which straight lines contain lattice points (points with whole-number coordinates). Every line (other than vertical ones) has a slope and an intercept. We can write its equation as MATH. Throughout, we will use the greek letters $\alpha $ and $\beta $ to denote irrational numbers. All other numbers will be whole numbers (e.g. $A,B,C,D$). The following table shows the four possibilities:


MATH

As usual, this should be written up neatly, carefully, and in complete readable sentences. Try to give reasons for all of your assertions.

  1. Explain why there is exactly one lattice point in the first case. (What is it, and why?)

  2. Explain why there can be no lattice points on the line in the second case.

  3. In the third case, give examples of lines where there is a lattice point on the line, and where there can't be any. Justify your statements.

  4. Explain exactly what conditions determine the existence of a lattice point in the last case. Give specific examples of such lines which do and don't contain lattice points. (You should assume that $\gcd (A,B)=1$ and $\gcd (C,D)=1$.) If there is a lattice point, explain how to find all lattice points; give an example.

  5. Write a Matlab program which inputs the numbers $A,B,C,D$, and the size N of a square in the first quadrant, and then finds all the lattice points on the line MATH which lie in the square (MATH). (You'll need a double "for" loop; multiply the equation through by $BD$ to get rid of fractions.)

    Since all numbers are stored in a computer as a finite number of "digits", no number can be irrational as far as a computer is concerned. Thus, we can not use a computer to tell when a number is irrational or not, so we can not use it to tell when lines with irrational coefficients contain rational points.

    One final note. If a line has irrational slope $\alpha $, and it contains no lattice point, it can be proved that it "passes arbitrarily close" to lattice points. This means that, for any distance $d>0$ you may specify, no matter how small (but not $0$), there is some lattice point that is within distance $d$ of the line. This is not easy to show, and is sometimes proved in a "real analysis" course. If you are interested in a proof, see me.

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