Goldbach Conjecture Graphs

Here are a few graphs on the Goldbach conjecture.

The Goldbach conjecture states that every even integer greater than two can be written as the sum of two primes. One question we can ask is, for any given even integer n, what is the smallest prime p such that there exists a prime q such that n = p + q. We can define for every even integer p(n), a function that is equal to this smallest p, or equal to -1 in the case that no such p exists. The latter definition is in case the Goldbach conjecture is false. Here are some values:

n p(n) n p(n) n p(n) n p(n)
4 2 54 7 104 3 154 3
6 3 56 3 106 3 156 5
8 3 58 5 108 5 158 7
10 3 60 7 110 3 160 3
12 5 62 3 112 3 162 5
14 3 64 3 114 5 164 7
16 3 66 5 116 3 166 3
18 5 68 7 118 5 168 5
20 3 70 3 120 7 170 3
22 3 72 5 122 13 172 5
24 5 74 3 124 11 174 7
26 3 76 3 126 13 176 3
28 5 78 5 128 19 178 5
30 7 80 7 130 3 180 7
32 3 82 3 132 5 182 3
34 3 84 5 134 3 184 3
36 5 86 3 136 5 186 5
38 7 88 5 138 7 188 7
40 3 90 7 140 3 190 11
42 5 92 3 142 3 192 11
44 3 94 5 144 5 194 3
46 3 96 7 146 7 196 3
48 5 98 19 148 11 198 5
50 3 100 3 150 11 200 3
52 5 102 5 152 3 202 3

And this is what the above data looks like on an xy plot:

The graph shows that usually a small number like 3 or 5 is sufficent, but occasionally larger primes are required. This graph shows the values for p(n) for n up to 10000:

And up to n = 100000:

Now define m(n) = max_{i<n} p(n). That is, m(n) is the maximum value of p(i) that was needed so far from i=2 to i=n. We get the following:

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