Problem 4

Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?

Step 3 of 4: Show no nonzero integer multiple gives sine zero

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\sin(k\theta)=\frac{a_k}{5^k},\qquad a_{k+1}=6a_k-25a_{k-1},\qquad a_k\equiv4\pmod5\ (k\ge1)

Detailed analysis

For $k\ge1$ , write $\sin(k\theta)=a_k/5^k$ . The recurrence gives $a_{k+1}=6a_k-25a_{k-1}$ , with $a_0=0,a_1=4$ , so $a_{k+1}\equiv a_k\equiv4\pmod5$ . Thus $\sin(k\theta)\ne0$ for $k\ne0$ ; negative $k$ follows from oddness.