# PRIMECOIN DISCOVERIES

The Primecoin is the first cryptocurrency with an additional scientific value derived from proof-of-work energy consumption. Each verified Primecoin block is also a numerical discovery of a prime number chain that can be useful to mathematicians. More than three years' worth of discoveries used to be hidden in blockchain.

PRIMES.ZONE is designed to serve as an online database of primecoin findings: of Cunningham chains of the 1st and 2nd kind, and of Bi-twin prime chains.

32 794 894 prime numbers in 3 192 480 chains have been found since July 2013.

## Mission

The idea of Primecoin is to use computational resources involved in blockchain verification for scientific purposes. There used to be web sites wherein one could browse all findings about prime number chains, but for the last few years, there has been no such site and the only way to obtain primecoin findings is to derive them directly from block metadata using a debugging console. Thus, without deep knowledge of blockchain technology, more than two million prime chains, including two world records, were inaccessible to potentially interested mathematicians.

The mission of the present web page is to make this prime number data easily accessible to everybody again.

If you think this site is useful for cryptocurrency and scientific community, please donate. Donations will help to keep this page running.

## Mathematical background

A Cunningham chain of the first kind of length $$k$$ is an array of prime numbers:
$$n-1, 2n-1, ..., 2^{k-1}n-1$$.
Number $$n$$ is called the origin of prime chain.
Example
- origin: $$n=3$$
- length: $$k=5$$
- chain: $$2,5,11,23,47$$

A Cunningham chain of the second kind of length $$k$$ is an array of prime numbers:
$$n+1, 2n+1, ..., 2^{k-1}n+1$$.
Number $$n$$ is called the origin of prime chain.
Example
- origin: $$n=18$$
- length: $$k=3$$
- chain: $$19, 37, 73$$

A Bi-twin prime chain of length $$2k$$ is chain of $$k$$ twin primes:
$$n-1, n+1, 2n-1, 2n+1, ...$$
$$..., 2^{k-1}n-1, 2^{k-1}n+1$$
Example
- origin: $$n=6$$
- length: $$2k=4 (k=2)$$
- chain: $$5, 7, 11, 13$$
Bi-twin prime chains can also be viewed as union of two Cunningham chains of the first and second kinds with same origin and length.

Primecoin proof-of-work algorithm is searching for prime chains with given length such that origin of the chain is divisible by the block header hash. Consequently, chain origin and primes are larger than the header hash (256 binary / 78 decimal digits).

Each verified block in primecoin blockchain corresponds to a prime chain. For more technical details see primecoin white paper. Check wikipedia pages for further reading about prime chains:

## Browse prime chains

Use buttons to browse trough blocks or type block id in edit box.

For more (financial) details about block follow the link by clicking the block header hash. Prime numbers in the table are links into Wolfram Alpha as an independent source to confirm the primality.

## World records

You can click on the buttons inside the tabbed menu to see largest prime numbers found by primecoin for each chain type and chain length.

Some of these finding are world records. For more details see Wikipedia pages: Cunningham chain, Bi-twin chain.

Cunningham chains of the 1st kind
7 244 4346
8 167 9021
9 151 57284
10 146 182690
11 140 95569
12 113 558800
13 107 368051
14 98 2931328
Cunningham chains of the 2st kind
6 134 54
7 208 17715
8 183 26660
9 167 79349
10 145 519253
11 127 365304
12 109 323183
13 101 539977
14 100 547276
Bi-twin prime chains
6 97 46
7 268 7101
8 172 11900
9 163 75899
10 138 479357
11 124 487155
12 118 476538
13 105 340499
14 100 3117396
15 98 2908166

## Visualisation

The visualisation of prime chains is very difficult, because they are rare and distributed over several powers of ten. To get a visual notion of just how many prime numbers have been found and how large the search space is, please take a look at the following visualisation.

The 64-mega-pixel image map represents our visualisation of the prime numbers found by primecoin: 32 794 894 prime numbers in 3 192 480 chains. Each pixel corresponds to one or more prime numbers.