Voronoi Diagrams of the Ulam Prime Spiral


The table below shows the primes that have Voronoi cells with perimeter larger than any smaller prime. I conjecture that this sequence is an infinite sequence - that is, there's no 'largest' 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
2(1,0)5
(3

2
,1

2
)(3

2
,-3

2
)(1

6
,-5

6
)(-1

4
,0)(0,1

2
)
61

24
7

2
+4

3
5
3(1,1)5
(0,1

2
)(3

2
,1

2
)(2,1)(1

2
,5

2
)(0,7

3
)
65

24
10

3
+22+1

6
10
5(-1,1)6
(-2,1)(-1,0)(-1

4
,0)(0,1

2
)(0,7

3
)(-1

2
,5

2
)
157

48
31

12
+5

2
2+1

4
5+1

6
10
11(2,0)5
(3

2
,1

2
)(3

2
,-3

2
)(3,-2)(3,0)(2,1)
7

2
4+3

2
2+1

2
10
23(0,-2)4
(1

6
,-5

6
)(-3

2
,-5

2
)(-1,-4)(3

2
,-3

2
)
25

6
25

6
2+2

3
5+1

2
10
47(1,-3)4
(-1,-4)(3

2
,-3

2
)(3,-2)(1

2
,-9

2
)
5
52+10
79(2,-4)4
(1

2
,-9

2
)(3,-2)(15

4
,-17

4
)(2,-6)
51

8
23

4
2+3

4
10
83(5,-3)5
(6,-5)(6,-5

3
)(5,-4

3
)(3,-2)(15

4
,-17

4
)
95

12
10

3
+5

2
10
137(2,6)5
(1,5)(1,8)(4,7)(14

3
,19

3
)(4,5)
49

6
6+2

3
2+2

3
5+10
311(5,9)7
(4,11)(4,7)(14

3
,19

3
)(6,7)(20

3
,9)(13

2
,19

2
)(5,11)
37

4
5+13

6
2+2

3
5+5

6
10
337(-9,-3)5
(-10,-5)(-10,-3

5
)(-9,-2

5
)(-17

2
,-1

2
)(-7,-2)
79

10
22

5
+9

2
2+3

10
26
443(11,-9)6
(9,-8)(10,-11)(12,-11)(51

4
,-35

4
)(32

3
,-20

3
)(10,-7)
259

24
2+37

12
2+1

3
5+7

4
10
673(-9,13)5
(-41

4
,47

4
)(-11,14)(-7,46

3
)(-47

7
,104

7
)(-8,11)
311

28
173

42
10+2

21
34
919(-15,-3)7
(-16,-5)(-121

7
,-8

7
)(-188

11
,-9

11
)(-31

2
,-1

2
)(-53

4
,-11

4
)(-57

4
,-23

4
)(-44

3
,-17

3
)
23641

1848
9

4
2+2

3
5+16

7
10+53

132
26+5

77
34
1993(-10,-22)6
(-8,-21)(-19

2
,-51

2
)(-89

7
,-171

7
)(-13,-24)(-13,-22)(-11,-20)
519

28
2+22+25

7
10+1

7
13
2719(-26,12)7
(-25,10)(-47

2
,29

2
)(-197

8
,119

8
)(-82

3
,43

3
)(-86

3
,35

3
)(-173

6
,65

6
)(-53

2
,17

2
)
983

48
23

6
2+4

3
5+15

8
10+17

24
26
3911(-27,-31)7
(-29,-30)(-29,-34)(-28,-104

3
)(-55

2
,-69

2
)(-24,-31)(-24,-29)(-28,-29)
83

4
10+9

2
2+1

6
10+1

3
13
7433(-43,7)8
(-44,4)(-81

2
,15

2
)(-163

4
,35

4
)(-85

2
,21

2
)(-91

2
,15

2
)(-367

8
,45

8
)(-181

4
,15

4
)(-134

3
,11

3
)
329

16
26

3
2+1

3
5+5

8
10+5

8
26
7621(44,8)8
(42,7)(122

3
,29

3
)(127

3
,34

3
)(87

2
,23

2
)(93

2
,17

2
)(234

5
,7)(369

8
,47

8
)(177

4
,19

4
)
2941

120
31

4
2+4

3
5+3

10
26+3

5
34
8867(-47,17)6
(-50,19)(-224

5
,19)(-259

6
,293

18
)(-1017

23
,318

23
)(-93

2
,27

2
)(-99

2
,33

2
)
27668

1035
26

5
+213

46
2+1

2
26+49

90
34+145

414
58