If you sum up an infinite series of 1/N^a, where a is greater than 1, the sum is finite. The following is an estimate of how many XPM will be produced given some assumptions.

XPM is produced at a fixed rate (one block per minite) and the number of XPMs per block (XPB) is reducing as a function of difficulty (diff), i.e.

XPB = 999 / diff^2

Although diff changes all the time, very months there are XPM primecoins mined on average,

XPM = XPB * (365/12*24*60) .

Suppose there were N0 of computation power (think total “hash power”) that could find 1 BPM when primecoin was released, at diffifulty=6. Assuming the increase of computation power, N, is only due to Moore’s Law, wich says N will double every 18 months. i.e.

N / N0 = 2^(t/18),

where t is time (in unit of months) since the release of XPM.

Assuming there are 10 times less target chains every time difficulty increases by 1, and for the same amount of N there are N*c 6 chains to befount, where c is a constant of 6 chain rate found per unit of N, then we have

1 BPM = N0 * c = N *c * 10^-(diff-6)

i.e.

N / N0 = 10^(diff - 6)

We can see that

diff = 6 + LOG2 * (t/18)

So the number of XPM mined everymonth can be calculated as a function of time. I didn’t borther to do an analytical integration. Instead I put the above in an spreadsheet. The following shows time (months since release), network computation power relative to release time, difficulty, XPMs per block, XPMs mined per months, and total number of XPMs, for the next 15 years, and at 30, 150 years after release.

```
t N/N0 diff XPB XPM SUM
0 1.00 6.00 27.75 1215450.00 1215450.00
1 1.04 6.30 25.16 1102088.59 2317538.59
2 1.08 6.60 22.92 1003877.37 3321415.96
3 1.12 6.90 20.96 918232.22 4239648.18
4 1.17 7.20 19.25 843097.35 5082745.53
5 1.21 7.51 17.74 776820.80 5859566.34
6 1.26 7.81 16.39 718062.88 6577629.22
7 1.31 8.11 15.20 665727.90 7243357.12
8 1.36 8.41 14.13 618912.71 7862269.83
9 1.41 8.71 13.17 576867.53 8439137.36
10 1.47 9.01 12.31 538965.67 8978103.03
11 1.53 9.31 11.52 504680.08 9482783.11
12 1.59 9.61 10.81 473564.95 9956348.06
13 1.65 9.91 10.17 445241.07 10401589.13
14 1.71 10.21 9.57 419384.31 10820973.44
15 1.78 10.52 9.03 395716.24 11216689.69
16 1.85 10.82 8.54 373996.64 11590686.33
17 1.92 11.12 8.08 354017.35 11944703.68
18 2.00 11.42 7.66 335597.29 12280300.97
19 2.08 11.72 7.27 318578.34 12598879.31
20 2.16 12.02 6.91 302821.92 12901701.24
21 2.24 12.32 6.58 288206.17 13189907.41
22 2.33 12.62 6.27 274623.59 13464531.00
23 2.42 12.92 5.98 261979.04 13726510.03
24 2.52 13.22 5.71 250188.09 13976698.13
25 2.62 13.53 5.46 239175.61 14215873.74
26 2.72 13.83 5.23 228874.55 14444748.29
27 2.83 14.13 5.01 219224.92 14663973.20
28 2.94 14.43 4.80 210172.93 14874146.13
29 3.05 14.73 4.60 201670.22 15075816.35
30 3.17 15.03 4.42 193673.25 15269489.60
31 3.30 15.33 4.25 186142.68 15455632.28
32 3.43 15.63 4.09 179042.93 15634675.20
33 3.56 15.93 3.93 172341.76 15807016.97
34 3.70 16.24 3.79 166009.89 15973026.85
35 3.85 16.54 3.65 160020.66 16133047.51
36 4.00 16.84 3.52 154349.79 16287397.30
37 4.16E+00 17.14 3.40 148975.12 16436372.43
38 4.32E+00 17.44 3.28 143876.37 16580248.80
39 4.49E+00 17.74 3.17 139034.97 16719283.77
40 4.67E+00 18.04 3.07 134433.89 16853717.66
41 4.85E+00 18.34 2.97 130057.48 16983775.14
42 5.04E+00 18.64 2.87 125891.35 17109666.49
43 5.24E+00 18.94 2.78 121922.24 17231588.74
44 5.44E+00 19.25 2.70 118137.92 17349726.66
45 5.66E+00 19.55 2.61 114527.10 17464253.76
46 5.88E+00 19.85 2.54 111079.33 17575333.09
47 6.11E+00 20.15 2.46 107784.93 17683118.02
48 6.35E+00 20.45 2.39 104634.95 17787752.97
49 6.60E+00 20.75 2.32 101621.06 17889374.03
50 6.86E+00 21.05 2.25 98735.54 17988109.57
51 7.13E+00 21.35 2.19 95971.20 18084080.77
52 7.41E+00 21.65 2.13 93321.35 18177402.12
53 7.70E+00 21.95 2.07 90779.74 18268181.86
54 8.00E+00 22.26 2.02 88340.57 18356522.43
55 8.31E+00 22.56 1.96 85998.41 18442520.84
56 8.64E+00 22.86 1.91 83748.17 18526269.01
57 8.98E+00 23.16 1.86 81585.11 18607854.11
58 9.33E+00 23.46 1.82 79504.78 18687358.89
59 9.70E+00 23.76 1.77 77503.01 18764861.90
60 1.01E+01 24.06 1.73 75575.91 18840437.81
15yr 1.02E+03 60.19 0.28 12079.73 22434618.42
30yr 1.05E+06 114.37 0.08 3345.10 23574469.88
150yr 1.27E+30 547.85 0.00 145.78 24578464.36
```

The difficulty doubles every 18 month as expected. The per block return will be less than 0.1XPM after 30 years. The total number of primecoins ever mined will be not much more than 25 million, comparable to that of bitcoins.

We have made some assumptions above. Since primecoin was released, N has been increasing much faster than Moore’s Law becuause more miners joins in, and better algorithms have been developed. Modifying Moore’s Law to N / N0 = 3^(t/18) gives a total cap of ~15 million XPMs. However there is no guarantee that N will deviate from Moore’s Law in long terms.

Mikaelh has found out that the ratio of 9 chains to 10 chains is not 10, but more close to 30. If every time difficulty increases by 1 there will be less than 1/10 times target chains, the effect is similar decreasing the number “2” in Moore’s Law, increasing the total sum.

To summarize, with reasonable assumptions there will be a cap of primecoins, of about 25 million.

Edit: wrong lable of years in table and typo. Add the missing first “-” in N *c * 10^-(diff-6). Changed “greater than 2” to “…1” in the first line.