Give an equation for a line that contains no lattice points

Give an equation for a line that contains no lattice points. Explain how you know it contains no lattice points.

Solution

Assuming you mean the lattice of integers, that is equivalent to checking if the equation has integer solutions.

Assume that k×GCD(A,B)=Ck×GCD(A,B)=C. Then by Euclid\'s algorithm, there are X,YX,Y such that AX+BY=GCD(A,B)AX+BY=GCD(A,B), so AkX+BkY=CAkX+BkY=C, where kXkX and kYkY must be integers.

On the other hand, assume Ax+By=CAx+By=C has a solution. Then GCD(A,B)(AGCD(A,B)X+BGCD(A,B)Y)=CGCD(A,B)(AGCD(A,B)X+BGCD(A,B)Y)=C so GCD(A,B)GCD(A,B) divides C.