(From I. M. Gelfand) If it is known that the straight line y = kx + b passes through two integral points, are there any other integral point on this straight line?
I tried out contradiction. Suppose (x1, y1) and (x2, y2) are the two integral points through which the given line passes. And further, that there are no more integral points on this line. So, any other point, say (x3, y3) on the line cannot be integral.
I have three equations: y1 = kx1 + b; y2 = kx2 + b; y3 = kx3 + b
and can get to a relation between (x1, y1), (x2, y2) and (x3, y3) but this looks like a long drawn process. Any Hint?
change $x$ by $x_2-x_1$ changes the value of $y$ by an integer so just add $x_2-x_1$ to $x_2$ to get another integer. There are therefore an infinite number of integral points.