Formulation of Somsky Gears as Diophantine Problem ========Warren D Smith====July 2015=============== Suppose the sun gear has S teeth, the outer ring gear has R teeth, the K planets have P1, P2, ..., PK teeth, and the distance between the sun-center and ring-center is equivalent to the radius of a gear with D teeth. Those involve K+2 positive-integer-valued variables S, R, P1, P2, ..., PK. Here D>0 need not be integer, it might merely be real. For each planet whose center is not collinear with SunCenter and RingCenter, it is possible to adjoin a "twin" planet reflected about the SunCenter-RingCenter line. I will ignore those twin planets, focusing here only on primal planets, all of which lie on same side of that line. Then by the law of cosines: SunCenter-RingCenter angle viewed from planetJ = arccos [D^2 - (PJ+S)^2 - (R-PJ)^2]/[2*(PJ+S)*(R-PJ)] RingCenter-PlanetJ angle as viewed from SunCenter = arccos [(R-PJ)^2 - (PJ+S)^2 - D^2]/[2*(PJ+S)*D] SunCenter-PlanetJ angle as viewed from RingCenter = arccos [(PJ+S)^2 - (R-PJ)^2 - D^2]/[2*(R-PJ)*D] Now the variously named "belt condition" or "alternating sum condition" is that the following quantity (regardless of the indices A and B with 0<A<B<=K) must be a rational multiple of pi. These arise from considering closed "belts" always contacting either sun, planetA, ring, or planetB, and traversing them in the turning direction. [The circumference of any such belt is demanded to be an integer number of teeth.] R*arccos [(PA+S)^2 - (R-PA)^2 - D^2]/[2*(R-PA)*D] - R*arccos [(PB+S)^2 - (R-PB)^2 - D^2]/[2*(R-PB)*D] + S*arccos [(R-PA)^2 - (PA+S)^2 - D^2]/[2*(PA+S)*D] - S*arccos [(R-PB)^2 - (PB+S)^2 - D^2]/[2*(PB+S)*D] - PA*arccos [D^2 - (PA+S)^2 - (R-PA)^2]/[2*(PA+S)*(R-PA)] - PB*arccos [D^2 - (PB+S)^2 - (R-PB)^2]/[2*(PB+S)*(R-PB)] If these rationality conditions are always satisfied (and if I did not screw up the signs in the first column; correct those if necessary) then the given S,R,P1,P2,...,PK work (at least if scaled up by some suitable integer multiple) as Somsky gear-tooth counts. It should suffice if these rationality conditions merely hold when A=1; the rationality of the others would then follow automatically. If so there are K-1 quantities which need simultaneously to be rational. To assure this we need to adjust K+2 integer quantities, plus one continuously variable quantity D. It is possible to rephrase the above in a way that avoids use of arctrig and trig functions, instead employing integer powers of complex numbers, and square root function, only. That makes it clear that D is always going to be an ALGEBRAIC number.