### 11th Putnam 1951

**Problem A5**

Show that a line in the plane with rational slope contains either no lattice points or an infinite number. Show that given any line L of rational slope we can find δ > 0, such that no lattice point is a distance k from L where 0 < k < δ.

**Solution**

The first part is obvious. If the line passes through (m, n) and has slope a/b (where m, n, a, b are all integers), then it also passes through (m + kb, n + ka) for any integer a. Although not required, we note that there are lines with rational slope passing through no lattice points, for example, y = 1/2 and y = x + 1/2.

Suppose L is bx - ay + c = 0. The distance of L from (x, y) is |bx - ay + c|/√(a^{2} + b^{2}). If (x, y) is a lattice point then bx - ay is an integer. Hence |bx - ay + c| is either zero or at least 1 if c is an integer, or at least the distance of c from the nearest integer if c is not an integer. So we can find k > 0, so that |bx - ay + c| > k for all lattice points not on L. Hence the distance |bx - ay + c|//√(a^{2} + b^{2}) is at least k/√(a^{2} + b^{2}) for all lattice points not on L. So we can take δ to be k/√(a^{2} + b^{2}).

11th Putnam 1951

© John Scholes

jscholes@kalva.demon.co.uk

5 Mar 2002