Primes in Apollonian Circle Packings. Primitive curvatures For any generating curvatures (sum is as...
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Transcript of Primes in Apollonian Circle Packings. Primitive curvatures For any generating curvatures (sum is as...
![Page 1: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/1.jpg)
Primes in Apollonian Circle Packings
![Page 2: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/2.jpg)
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)
![Page 3: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/3.jpg)
Parity of Mutually Tangent Circles
• All groups of four mutually tangent circles in primitive curvatures have two even and two odd curvatures.
![Page 4: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/4.jpg)
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
![Page 5: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/5.jpg)
Curvatures vs Prime Curvatures: (-1,2,2,3) x/log(x)
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![Page 6: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/6.jpg)
Curvatures vs Prime Curvatures : (0, 0, 1, 1)
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Curvatures vs Prime Curvatures : Or (-12,25,25,28)
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![Page 8: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/8.jpg)
Curvatures vs Prime Curvatures : (-6,10,15,19)
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![Page 9: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/9.jpg)
Curvatures vs Prime Curvatures : (-4, 8, 9, 9)
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![Page 10: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/10.jpg)
For Integers vs Prime Integersx/log(x)
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![Page 11: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/11.jpg)
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.
![Page 12: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/12.jpg)
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”.
![Page 13: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/13.jpg)
Curvatures: (-1,2,2,3)Mod 2
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![Page 14: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/14.jpg)
Curvatures (-1,2,2,3)Mod 3
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![Page 15: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/15.jpg)
Curvatures: (-1,2,2,3)Mod 24
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![Page 16: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/16.jpg)
Curvatures (-1,2,2,3)Mod 7
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![Page 17: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/17.jpg)
Curvatures (-1,2,2,3)Mod 13
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![Page 18: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/18.jpg)
Curvatures (0,0,1,1)Mod 2
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![Page 19: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/19.jpg)
Curvatures (0,0,1,1)Mod 3
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![Page 20: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/20.jpg)
Curvatures (0,0,1,1)Mod 24
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![Page 21: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/21.jpg)
Curvatures (0,0,1,1)Mod 7
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![Page 22: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/22.jpg)
Curvatures (0,0,1,1)Mod 13
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![Page 23: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/23.jpg)
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…
![Page 24: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/24.jpg)
Zeros are numbers that do occur.In generation 2, we have…
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![Page 25: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/25.jpg)
In generation 6
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![Page 26: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/26.jpg)
In Generation 10
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![Page 27: Primes in Apollonian Circle Packings. 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.](https://reader030.fdocuments.us/reader030/viewer/2022032800/56649d2c5503460f94a02415/html5/thumbnails/27.jpg)
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.