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Primes in Apollonian Circle Packings
Primitive curvatures
• For any generating curvatures (sum is as small as possible under Si ) 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)
Parity of Mutually Tangent Circles
• All groups of four mutually tangent circles in primitive curvatures have two even and two odd curvatures.
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
Curvatures vs Prime Curvatures: (-1,2,2,3) x/log(x)
0 2000 4000 6000 8000 10000 12000 140000
200
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1400
1600
Curvatures vs Prime Curvatures : (0, 0, 1, 1)
x/log(x)
0 0.5 1 1.5 2 2.5 3 3.5
x 104
0
500
1000
1500
2000
2500
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3500
Curvatures vs Prime Curvatures : Or (-12,25,25,28)
x/log(x)
0 2000 4000 6000 8000 10000 12000 140000
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1400
Curvatures vs Prime Curvatures : (-6,10,15,19)
x/log(x)
0 2000 4000 6000 8000 10000 12000 140000
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1000
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1400
Curvatures vs Prime Curvatures : (-4, 8, 9, 9)
x/log(x)
0 2000 4000 6000 8000 10000 12000 140000
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1400
For Integers vs Prime Integersx/log(x)
0 2000 4000 6000 8000 10000 12000 140000
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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.
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”.
Curvatures: (-1,2,2,3)Mod 2
0 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
4
mod n
how
man
y cu
rvat
ures
for
eac
h m
od
Curvatures (-1,2,2,3)Mod 3
0 1 20
0.5
1
1.5
2
2.5x 10
4
Curvatures: (-1,2,2,3)Mod 24
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 230
1000
2000
3000
4000
5000
6000
7000
Curvatures (-1,2,2,3)Mod 7
0 1 2 3 4 5 60
1000
2000
3000
4000
5000
6000
7000
Curvatures (-1,2,2,3)Mod 13
0 1 2 3 4 5 6 7 8 9 10 11 120
500
1000
1500
2000
2500
3000
3500
Curvatures (0,0,1,1)Mod 2
0 10
2000
4000
6000
8000
10000
12000
14000
16000
18000
Curvatures (0,0,1,1)Mod 3
0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
4
Curvatures (0,0,1,1)Mod 24
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 230
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Curvatures (0,0,1,1)Mod 7
0 1 2 3 4 5 60
1000
2000
3000
4000
5000
6000
Curvatures (0,0,1,1)Mod 13
0 1 2 3 4 5 6 7 8 9 10 11 120
500
1000
1500
2000
2500
3000
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…
Zeros are numbers that do occur.In generation 2, we have…
0 10
200
400
600
800
1000
1200
1400
1600
1800
In generation 6
0 10
200
400
600
800
1000
1200
1400
1600
1800
In Generation 10
0 10
100
200
300
400
500
600
700
800
900
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.
