On a Diophantine Equation That Generates All Integral Apollonian Gaskets

Algorithm 1

An algorithm that produces all integral disk packings.

for 𝐵=1 to whatever you wish

  for 𝜇=0 to int (𝐵/√3)

      evaluate 𝐻=𝐵²+𝜇²

   for 𝑘=2𝜇 to int(√𝐻)

       if 𝑛=𝐻/𝑘 is an integer and gcd (𝐵,𝑘,𝑛)=1

       then the quintet of main curvatures is

    (−𝐵,𝐵+𝑘,𝐵+𝑛,𝐵+𝑘+𝑛−2𝜇,𝐵+𝑘+𝑛+2𝜇)

  next 𝑘

 next 𝜇

next 𝐵