login
The curvature of smallest circle among 4 mutually tangent(externally) circles with integer curvature and primitive (share no common factor).
0

%I #22 Jul 10 2016 22:59:32

%S 12,15,23,24,28,33,34,35,38,39,40,42,45,47,50,52,53,56,57,58,59,60,61,

%T 62,63,63,64,66,69,71,72,72,73,76,77,77,79,80,81,82,82,83,84,84,85,86,

%U 87,87,88,90,91,91,94,94,95,95,96,96,97,98,98,99,99

%N The curvature of smallest circle among 4 mutually tangent(externally) circles with integer curvature and primitive (share no common factor).

%C 4 mutually tangent circles satisfy 2 (a^2 + b^2 + c^2 + d^2) = (a + b + c + d)^2 where a,b,c,d are the curvatures.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Apollonian_gasket">Apollonian gasket</a>

%e a, b, c, d

%e 12, 4, 1, 1

%e 15, 3, 2, 2

%e 23, 6, 3, 2

%e 24, 12, 1, 1

%e 28, 9, 4, 1

%t aMax = 100;

%t Do[

%t If[GCD[a, b, c] > 1, Continue[]];

%t d = a + b + c - 2 Sqrt[a b + a c + b c];

%t If[d // IntegerQ // Not, Continue[]];

%t (*{a,b,c,d}*)a // Sow;

%t , {a, aMax}

%t , {b, (2 a)/Sqrt[3] - a // Ceiling, (Sqrt[a] - 1)^2}

%t , {c,(a-b)^2/(4(a+b))//Ceiling,Min[b,(Sqrt[a]-Sqrt[b])^2-1//Ceiling]}

%t ] // Reap // Last // Last(*//TableForm*)

%t d =.;

%K nonn

%O 1,1

%A _Albert Lau_, Jul 03 2016