Dickson's "Introduction to the Theory of Numbers" is mostly about Diophantine Equations. There's a chapter on ax2+by2+cz2=0. I think it's called Legendre's Equation. I have the Dover reprint, but can't put my hands on it at the moment. Hardy & Wright have a tiny DEq chapter. They cover the vanilla x2+y2=z2, and might stretch to include one coefficient. The typical situation is a small number of parametric solutions similar to the Pythagorean case (x2+y2=z2). I think the rational case can be written with just one parameter, not sure. The Pyth case can arrange all primitive solutions in a nice ternary tree, starting from 345 (or 011). The children are x' = 2x+y+2z, y' = x+2y+2z, z' = 2x+2y+3z. Varying the signs of x & y takes you in four directions: 3 children, one parent. Something similar probably exists for your modification (*). This arrangement of the variables gives a nice ordering of the tree by atan(x/y). If you swap the defns of x' & y', then you can preserve even-odd parity in the solutions, useful for your 4x2 term. Rich -------- Quoting Fred Lunnon <fred.lunnon@gmail.com>:
I hunted the "octocog" with 6-fold symmetry: like the snark, it remains elusive.
Such a gear train would comprise a sun gear centred at the origin, of radius s ; which meshes with three counter-rotating inner planets of radius q ; which each mesh also with two of three outer planets of radius p ; which all mesh with an external ring of radius r .
The constraint that corresponding discs should touch reduces to 4*X^2 - 3*Y^2 = W^2 (*) where X == p-q , Y == s-q , Z == r-p , W = 2 Z - Y . The meshing constraints involve belts around discs s,q,p,q and r,p,q,p resp.
Essentially the only solutions found for X < 1024 are scalar dilations of [X, Y, Z, W] = 12 [1, 1, 1, 1] corresponding to concentric increments of the degenerate [r, s, p, q] = [18, 6, 6, -6] --- six equal planets touching both a sun and a ring of triple radius. Any increment of this has overlapping discs.
The diophantine tangency equation (*) has some independent interest. Apparently a complete set of primitive, non-negative solutions is generated (after reducing by GCD and eliminating repetitions) by the set of three parametric forms [X, Y, Z, W] , [X, Z, Y, X+Y-W] , [X, |Y-Z|, Y, X+Y] (**) where X = u^2 + 3*v^2 , Y = 4*u*v , W = 2*|u^2 - 3*v^2| , Z = (Y+W)/2 .
The first few primitive solutions are 1, 0, 1, 2 1, 1, 1, 1 7, 3, 8, 13 7, 5, 8, 11 7, 8, 5, 2 13, 7, 15, 23 13, 8, 15, 22 13, 15, 8, 1 19, 5, 21, 37 19, 16, 21, 26 19, 21, 16, 11 31, 11, 35, 59 31, 24, 35, 46 31, 35, 24, 13
This is a special case of the natural extension of Pythagorean triples to solutions of (say) a X^2 = b Y^2 + c Z^2 ; however I could not locate any relevant discussion --- references, anyone?
Fred Lunnon
On 10/19/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
This problem impinged during my customary brainstorming session in search of convincing reasons not to get out of bed. I have not examined it, so have no idea whether it is well-known, easy, hard or impossible.
Deventer's "Gear Shift" puzzle http://www.jaapsch.net/puzzles/gearshift.htm is topologically equivalent to an octahedron with a gear on each face, meshing with three gears adjacent at edges.
Is it possible to morph this concept into its planar map: a flat train comprising one ring enclosing two 3-planet tiers around one sun?
Furthermore, do such trains exist with 6-fold, 2-fold, and no symmetry?
Fred Lunnon
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