Primecoin long term money supply and total cap

Marketplace Primecoin
1

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 * (365/122460) .

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.

2

XPM has incredible beauty. If there is a coin to go religious on, for me (as a mathematics enthusiast), it is XPM.

3

It’s a candidate for inter-galactic commerce. Every block can be traced to a Cunningham chain which is both unique and universal no matter where or in which dimension you live It’s like the serial number printed on the bill. Now the inter-galactic blockchain is going to be big… :smiley:

4

After a solid year of researching bitcoin, I may now be ready to begin to understand XPM. For reference, I have a math bachelors and a physics masters. This isn’t to gloat, but rather a tip of my hat to sunny because his is the only whitepaper I cannot fucking understand.

I have too many questions.

5

as someone who did maths as well… i find most of this beyond me as well and have so many questions I would love answered, and yup juat makes me take my hat for sunny at the sheer genius of his mind and abilities :slight_smile:

Fuzzybear

6

Yep…I find myself teaching bitcointalk users about prime characteristic fields, you know, the easy shit.

XPM circlejerk here - this is some SERIOUS number theory…and, like. just wow.

I recently wrote to a reddit user on peercoin who was hating on XPM:

[size=8pt]"XPM has unique, remarkable properties. For example, there is no hard coded limit in the supply yet it is guided by the theorems we know from number theory about the distribution of the prime numbers. From this we can easily infer some reasonable upper bounds in the 20 milliion range for the next century or so - that is, barring some paradigm shift in computing. But the interesting thing about THAT is that XPM will respond to such a paradigm shift, as if to take on whatever form it needs to in order to compliment the nature of the computing world it lives in.

In essence, moore’s law itself governs the coin cap, which is absolutely distinct economically from all other cryptocoins with hard coded limits. XPM seems as if to “breathe with nature” if you catch my drift.

XPM is my favorite coin in fact because of this. Its economics are somehow “natural” rather than arbitrary.
Moreover, I think you are missing a piece of the puzzle if you do not see the manner in which XPM is everything that PPC is not. Given sunny’s mathematical gifts, I have given considerable thought to the interaction of the two currencies and firmly believe he designed the two to be harmonious acting as set theoretic compliments of one another."[/size]

This is when I realized I am a religious man afterall, and that religion be XPM, bitch.

7

The sum is finite as long as a is greater than 1. 1.01 will also do.

8

The sum is finite as long as a is greater than 1. 1.01 will also do.[/quote]

Yes, you are right. My math is getting rusty. So Primecoin is even more likely to have a limited total number.

9

One problem. I’ve only been seeing that computers are getting cheaper rather than faster. You can buy more processors and run a cluster, but most people will opt for cheaper, less processing power. That’s one problem which may or may not affect profitability and thus is linked to the possibility/probability a skyrocketing difficulty level.

There is evidence that Moore’s Law is about to be broken, so there will be no more advances onwards as they have already pushed down to 14 nanometers. However, it would be great to see a dual core PC (3 Ghz) + RAM + Mobo + Case for under $50. I have serious doubts to whether this can actually happen or not at the moment due to the costs of the raw inputs.

More awareness of primecoin mining may increase the number of workers, but this has approached the maximum in my opinion due to the convergence to electricity costs and evidence that unprofitable(profitable for operator) botnets allowing for increases in difficulty. There comes a point where it becomes unprofitable to mine, and therefore it is more reasonable that there will actually be a very large number of total primecoins, possibly in the billions (within a 25 years) with the 5 million or so being created every month.

I doubt many people would be mining if the reward was 0.1 XPM.

So the assumptions that have been made in the analysis appear to require the “ceteris paribus” assumption.

I believe it is more likely that we will see billions of primecoins produced in the time frame you have mentioned rather than millions. Although, I still believe we will come out ahead of the 5% inflation which dogecoin is pegged at.

The sustainability and amount of primecoins which will be produced in total is actually linked to the value of bitcoins. If the price is low, then it is very unlikely people will mine when there is a 0.1 XPM reward. If the price of bitcoin is high, then possibly a 0.1 XPM reward per block would be sufficient for miners. However, the value of bitcoin is constrained heavily due to the comparable precious metal “gold”. It has already reached this point which is devastating news.

Electricity costs are more likely to rise in the future as well as the globe shifts towards renewable energy, further denting profitability.

Maybe when there are a set of quantum computers mining, the scenario of 25 million primecoins might be possible, but then the centralisation of mining might possibly force people to lose confidence in the coin altogether. I can’t imagine 1,000 quantum computers mining when the rewards are a few cents an hour (0.1 XPM per block shared amongst 1,000).

144 primecoins per day, $3 per primecoin, $0.432 per miner => $0.02 per hour.

10

[quote=“lindatess, post:9, topic:1794”]One problem. I’ve only been seeing that computers are getting cheaper rather than faster. You can buy more processors and run a cluster, but most people will opt for cheaper, less processing power. That’s one problem which may or may not affect profitability and thus is linked to the possibility/probability a skyrocketing difficulty level.

There is evidence that Moore’s Law is about to be broken, so there will be no more advances onwards as they have already pushed down to 14 nanometers. However, it would be great to see a dual core PC (3 Ghz) + RAM + Mobo + Case for under $50. I have serious doubts to whether this can actually happen or not at the moment due to the costs of the raw inputs.[/quote]

I used to play with an Apple ][ that cost 2k USD and way slower than the processor of today’s throw-away digital watches. Making a cheaper machine is the goal but getting a faster machine at the same price is the result, or a side effect if you will.

More awareness of primecoin mining may increase the number of workers, but this has approached the maximum in my opinion due to the convergence to electricity costs and evidence that unprofitable(profitable for operator) botnets allowing for increases in difficulty.

Botnets are part of the ecosystem. They convert common computation power to XPM network “hash rate”, which in the long term follows how fast common home/office/personal computers are.

I think devices such as google glasses will eventually push the processing speed of common personal devices to something faster than today’s fastest GPUs to match the full need of human vision (full color 50 million pixels at 100 fps refresh rate, running on batteries). Primecoin has been resistant to GPUs. But I think it’s a matter of time before someone finds out how to use GPUs to mine XPMs, contributing to the increase of total network computation power by a factor of 1000 - 10000 (only 2-4 steps in terms of target chain length).

There comes a point where it becomes unprofitable to mine, and therefore it is more reasonable that there will actually be a very large number of total primecoins, possibly in the billions (within a 25 years) with the 5 million or so being created every month.

So far we are way ahead of the simple Moore’s Law I used in the calculation. It’s been 31 weeks since Primecoin was released and we have mined more than 15% of 25 million coins. We will see if difficulty reaches 24 in five years.

Maybe when there are a set of quantum computers mining, the scenario of 25 million primecoins might be possible, but then the centralisation of mining might possibly force people to lose confidence in the coin altogether. I can't imagine 1,000 quantum computers mining when the rewards are a few cents an hour (0.1 XPM per block shared amongst 1,000).

Someone used to claim the whole US would only need three computers, which at that time takes quite a few large rooms to install. 8)

11

I guess I agree with your response, and it is clear you can see that the average consumer is important.

The reason why I look at what the average consumer would purchase is because currently those are the most affordable solutions that currently have a faster return on investment. That’s why it’s more interesting to look at what an average person would purchase. Even at the moment it isn’t affordable to go out to build a cluster as the real return is zero after accounting for electricity costs.

If you already have the equipment, then you may recover some of the cost of the equipment that you purchased. This average of reasonable level of equipment is where you should be analysing, even if the average equipment becomes much faster.

Now I only use the 1,000 quantum computers example to show that there would be centralisation of mining as this is the only affordable solution as people would not be mining when they are getting less than 2 cents an hour. This will have to happen and will destroy confidence regardless.

If we had 1,000,000 quantum computers mining primecoin, then there would be

144 primecoins per day, $2 per primecoin ($288 to share amongst 1,000,000 workers)

$0.000288 per miner => $0.000012 / 0.0012 cents per hour.

At 1,000 miners this becomes affordable and allows each miner to receive 1.2 cents per hour.

Roughly most computers consume 2 cents an hour in electricity which is why I smoothed it back to 1,000. However, even the current workers is over 10,000 which means that the primecoin block reward can never reach zero otherwise, it becomes unaffordable. Also there is no transaction fee earned by the miners, it is merely destroyed.

Furthermore, due to this there is an infinite total supply primecoins. The assumption is incorrect that the reward per block approaches zero as time goes to infinity.

The fact is at the extremities, as the reward per block is constrained above zero. As time goes to infinity, total supply goes to infinity. it’s the mathematical property of primecoin that it can never be zero. Furthermore, the cost of electricity will constrain it so that it will never approach anything near 0.1 XPM per block, unless we have the centralisation of mining amongst less than 1,000 miners.

NB: I’m not sure where to put this.

The average computer also affects the average infected computer in a botnet. If the average consumer only buys cheaper computers with the same processing power, then the contribution of a botnet to the increasing primecoin difficulty is diminished. Botnets are required to push the XPM per block closer to a “true zero” as they have nothing to lose. Imagine if they infected 1,000 quantum computers, that would mean it’s profitable for XPM to approach zero, but it can never be zero, otherwise the botnet operator and everyone else legitimately mining earns nothing.

12

I am curious how accurate the estimates/projections in the first post, describing the difficulty, have been. It seems to me that difficulty had been fairly steady for a while, before the gpu miners started. It seems like difficulty doubling every 18 months seems unlikely due to primecoin’s unique pow and the competition of so many other coins being released.

Mostly, I am wondering what the release of gpu miners has done to the estimates of total supply and how the op’s projections hold up thus far, over the life of primecoin.

Edit: i realized that doubling of difficulty is happening faster than 18 months because difficulty does not increase in a linear fashion and numerical value of difficulty is more complex than i thought. difficulty 12 is not 2x difficulty of 6. (sorry it has been a long time since I have used much maths, i am slow to remember…) :slight_smile:

13

Not very accurate in the first few years. /the difficulty is almost 11 now. In pure Moore’s law it should be less than 9.

Mostly, I am wondering what the release of gpu miners has done to the estimates of total supply and how the op's projections hold up thus far, over the life of primecoin.

In the above I mentioned if GPUs contribute “to the increase of total network computation power by a factor of 1000 - 10000” they increase difficulty by “only 2-4 steps”. Not a big deal in the long term. It’s a natural step.

14

Sorry only see your interesting post now.

I think you confused it with Peercoin. For Primecoin the tx fee goes to the miner.

The fact is at the extremities, as the reward per block is constrained above zero. As time goes to infinity, total supply goes to infinity.

The sum of an infinite numbers is not necessarily infinite. That is why I put the first sentence of OP. It’s all a matter of how slow/fast the reward decreases.

In the far distant future personal computational power is so greate that one person’s mobile device could push the block reward to 0.001XPM/block. If XPM is still worth the same in today’s USD, who would mine it? Will that cause the end of XPM and finite total supply? Then if that last person stops mining, the difficulty will drop like a rock and block reward would bounce back to 20XPM/block… We have seen such difficulty jumps in some of the coins already. I have not analyzed the long term effect of this on average difficulty. If it makes a<1 (in the first sentence of OP) then XPM will have an infinite supply.

15

infinite supply? How come?

Then if that last person stops mining, the difficulty will drop like a rock and block reward would bounce back to 20XPM/block…

so few miner can provide infinite supply?

16

[quote=“sabreiib, post:15, topic:1794”]infinite supply? How come?

Then if that last person stops mining, the difficulty will drop like a rock and block reward would bounce back to 20XPM/block…

so few miner can provide infinite supply?[/quote]

If every year a single miner mines 20XPM and stops, since there will be infinite number of years in the future, there will be infinite number of XPMs.

17

During my life, the XPM supply is finite. Small supply every year equals nothing.

In the long run, we are all dead.---- Keynes :))

18

[quote=“sabreiib, post:17, topic:1794”]During my life, the XPM supply is finite. Small supply every year equals nothing.

In the long run, we are all dead.---- Keynes :))[/quote]

That is fine. You limited life time has limited implication in this discussion however.

19

Mhps,according to
http://xpm.muuttuja.org/charts/

XPM supply Is steady at 6million per year, is that right?

20

Although I’m not mhps I might give you a reply :wink:
As the block space is approximately constant (at 1 block per minute), but the block reward decreasing (as a function of the steeply rising difficulty), there must be a growing supply (with a diminishing growth rate).
I admit I haven’t visited the link you posted, but the answer can be given by the XPM protocol :wink:

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