These are sums of cos(w*j), where j belongs to a set Ci and Di. Sets Ci and Di are of the form
{ x, M(4*x), M(16*x),..., M(16384)} and {M(2*x), M(8*x),..., M(32768)}
where x is the smallest element of Ai.
We split set Ci into two sets Ei and Fi of 4 elements each
Ei={ x, M(16*x), M(256*x), M(4096*x) }
Fi={ M(2*x), M(32*x), M(512*x), M(8192*x) }
We split set Di into two sets Gi and Hi of 4 elements each
Gi={ M(4*x), M(64*x), M(1024*x), M(16384*x) }
Hi={ M(8*x), M(128*x), M(2048*x), M(32768*x) }
Let us define
We have
In a known way, repeated many times, we can formulate equations with roots ei and fi. These equations are
shown on the applet below. Clicking label "Change" other equations appear roots of these new equations are
gi and hi.
In the following applet use applet's scrollbar to see whole formulas.
Now we are ready , for example, to derive value of cos(2*pi/65537).
Let us denote
Software that you really need.