# Voronoi Diagrams of the Ulam Prime Spiral

The table below shows the primes that have 'rounder' Voronoi cells than any smaller prime. I conjecture that this sequence is an infinite sequence - that is, there's no 'roundest' voronoi cell. If you aren't sure what all this means, check out the explanation on the main page on this topic.

p"Centre"#VerticesVertex ListAreaPerimeter(P/2)2/A
2(1,0)5
 ( 32 , 12 ) , ( 32 , - 32 ) , ( 16 , - 56 ) , ( - 14 , 0 ) , ( 0 , 12 )
 6124
 72 + 43 �5
4.132018689726583
3(1,1)5
 ( 0 , 12 ) , ( 32 , 12 ) , ( 2 , 1 ) , ( 12 , 52 ) , ( 0 , 73 )
 6524
 103 + 2 �2 + 16 �10
4.1298586647841296
5(-1,1)6
 ( - 2 , 1 ) , ( - 1 , 0 ) , ( - 14 , 0 ) , ( 0 , 12 ) , ( 0 , 73 ) , ( - 12 , 52 )
 15748
 3112 + 52 �2 + 14 �5 + 16 �10
3.967721553536328
37(-3,3)5
 ( - 4 , 3 ) , ( - 4 , 4 ) , ( - 3 , 5 ) , ( - 32 , 72 ) , ( - 3 , 2 )
 174
 1 + 5 �2
3.831890330807703
61(0,4)7
 ( 0 , 5 ) , ( - 32 , 72 ) , ( - 12 , 52 ) , ( 0 , 73 ) , ( 12 , 52 ) , ( 1 , 3 ) , ( 1 , 5 )
 5512
 3 + 3 �2 + 13 �10
3.7546790434997863
67(-4,2)8
 ( - 5 , 2 ) , ( - 5 , 1 ) , ( - 92 , 12 ) , ( - 4 , 13 ) , ( - 72 , 12 ) , ( - 3 , 1 ) , ( - 3 , 2 ) , ( - 4 , 3 )
 236
 2 + 3 �2 + 13 �10
3.4723249554225166
10243(51,-9)8
 ( 52 , - 9 ) , ( 1012 , - 212 ) , ( 3407 , - 697 ) , ( 3397 , - 647 ) , ( 49 , - 8 ) , ( 50 , - 7 ) , ( 51 , - 7 ) , ( 52 , - 8 )
 617
 2 + 72 �2 + 47 �5 + 914 �10 + 17 �26
3.4642618196721773
10243(51,-9)8
 ( 52 , - 9 ) , ( 1012 , - 212 ) , ( 3407 , - 697 ) , ( 3397 , - 647 ) , ( 49 , - 8 ) , ( 50 , - 7 ) , ( 51 , - 7 ) , ( 52 , - 8 )
 617
 2 + 72 �2 + 47 �5 + 914 �10 + 17 �26
3.4642618196721773
11087(53,9)10
 ( 51 , 10 ) , ( 51 , 353 ) , ( 52 , 12 ) , ( 1613 , 12 ) , ( 2234 , 434 ) , ( 4478 , 838 ) , ( 1663 , 233 ) , ( 54 , 7 ) , ( 1052 , 152 ) , ( 1032 , 172 )
 29916
 103 + �2 + 23 �5 + 3524 �10 + 1324 �26 + 512 �34
3.4425149109627373
15199(62,8)9
 ( 60 , 7 ) , ( 60 , 295 ) , ( 1843 , 5 ) , ( 63 , 5 ) , ( 1292 , 132 ) , ( 65 , 223 ) , ( 65 , 9 ) , ( 2554 , 414 ) , ( 2434 , 374 )
 2371120
 6815 + 114 �2 + 74 �10 + 1330 �34
3.4377179873404042
16763(65,57)3
 ( 66 , 55 ) , ( 3365 , 55 ) , ( 2033 , 1723 )
 10115
 65 + 53 �2 + 715 �26
1.3085199697006535
15199(62,8)9
 ( 60 , 7 ) , ( 60 , 295 ) , ( 1843 , 5 ) , ( 63 , 5 ) , ( 1292 , 132 ) , ( 65 , 223 ) , ( 65 , 9 ) , ( 2554 , 414 ) , ( 2434 , 374 )
 2371120
 6815 + 114 �2 + 74 �10 + 1330 �34
3.4377179873404042
16529(-48,-64)5
 ( - 50 , - 63 ) , ( - 49 , - 62 ) , ( - 46 , - 62 ) , ( - 2716 , - 3776 ) , ( - 1363 , - 1913 )
 12712
 3 + 116 �2 + 43 �5 + 16 �26
2.097909477057046
15199(62,8)9
 ( 60 , 7 ) , ( 60 , 295 ) , ( 1843 , 5 ) , ( 63 , 5 ) , ( 1292 , 132 ) , ( 65 , 223 ) , ( 65 , 9 ) , ( 2554 , 414 ) , ( 2434 , 374 )
 2371120
 6815 + 114 �2 + 74 �10 + 1330 �34
3.4377179873404042
17471(-66,20)9
 ( - 1292 , 412 ) , ( - 64 , 19 ) , ( - 65 , 18 ) , ( - 67 , 17 ) , ( - 3385 , 17 ) , ( - 69 , 583 ) , ( - 69 , 21 ) , ( - 67 , 22 ) , ( - 1312 , 432 )
 52130
 3415 + 2 �2 + 2 �5 + �10 + 715 �34
3.436491442936181