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.
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|/√(a2 + b2). 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|//√(a2 + b2) is at least k/√(a2 + b2) for all lattice points not on L. So we can take δ to be k/√(a2 + b2).
11th Putnam 1951
© John Scholes
5 Mar 2002