1. Primes in Apollonian Circle Packings

2. Primitive curvatures For any generating curvatures (sum is as small as possible under S i ) a,b,c,d then gcd(a,b,c,d)=1 If not there will be no primes beyond the first generation (i.e. these are not interesting to our project so we ignore them)

3. Parity of Mutually Tangent Circles All groups of four mutually tangent circles in primitive curvatures have two even and two odd curvatures.

4. Question : What is the ratio of prime curvatures to total curvatures? We wrote a program that plots the number of curvatures versus the number of prime curvatures in each generation. We compared the graphs of these plots up to the ninth generation for different root quadruples

5. Curvatures vs Prime Curvatures: (-1,2,2,3) x/log(x)

6. Curvatures vs Prime Curvatures : (0, 0, 1, 1) x/log(x)

7. Curvatures vs Prime Curvatures : Or (-12,25,25,28) x/log(x)

8. Curvatures vs Prime Curvatures : (-6,10,15,19) x/log(x)

9. Curvatures vs Prime Curvatures : (-4, 8, 9, 9) x/log(x)

10. For Integers vs Prime Integers x/log(x)

11. WHY? (Rough idea): If all integers can be written as the sum of four squares then all integers should show up in some circle packings If there is no “bias” in Apollonian circle packings, all packings should get roughly the same ratio of primes as all other packings and as the integers.

12. Modula n Which numbers mod n appear in the curvatures of a given generation? We wrote a program to look at which mods occur for each set of different curvatures. We also looked at “bad primes” and what made them “bad primes”.

13. Curvatures: (-1,2,2,3) Mod 2

14. Curvatures (-1,2,2,3) Mod 3

15. Curvatures: (-1,2,2,3) Mod 24

16. Curvatures (-1,2,2,3) Mod 7

17. Curvatures (-1,2,2,3) Mod 13

18. Curvatures (0,0,1,1) Mod 2

19. Curvatures (0,0,1,1) Mod 3

20. Curvatures (0,0,1,1) Mod 24

21. Curvatures (0,0,1,1) Mod 7

22. Curvatures (0,0,1,1) Mod 13

23. Does every m Mod n occur? We wrote a program to plot a histogram of those numbers of the form n mod m that do not occur versus those that occur. For 6 mod 24 with the packing (-1,2,2,3) and looking at numbers up to 10,000 we got…

24. Zeros are numbers that do occur. In generation 2, we have…

25. In generation 6

26. In Generation 10

27. WHY? (Rough idea): Local to global principles suggest that if some m mod n occurs somewhere in the packing then after local barriers are removed, all m mod n should occur.