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Block #31,651

2CCLength 8β˜…β˜†β˜†β˜†β˜†

Cunningham Chain of the Second Kind Β· Discovered 7/14/2013, 12:12:11 AM Β· Difficulty 7.9891 Β· 6,778,020 confirmations

2CC
Cunningham Chain of the Second Kind

A sequence where each prime is double the previous prime minus one.

Block Header
Block Hash
3b15bf30ffb2ae5a905a7c4bd8a2b442e3fca4ea0c413633585c72e6efd2d1a3
View on Chainz β†—

Height

#31,651

Difficulty

7.989065

Transactions

1

Size

199 B

Version

2

Bits

07fd335d

Nonce

1,171

Timestamp

7/14/2013, 12:12:11 AM

Confirmations

6,778,020

Merkle Root

85cff2c62b339c37726eec953b152b50a8f214fac2b904681f875311edf5c139
Transactions (1)
Coinbase85cff2c62b339c…875311edf5c139
1 in β†’ 1 out15.6500 XPM109 B
AKSwr9eU…dLNxuVyn
Prime Chain Origin

This is the prime chain origin stored in the block header. It is a composite number (not prime itself) β€” it equals the first prime in the chain multiplied by a primorial. The origin anchors the entire chain to this specific block.

3.585 Γ— 10⁹⁴(95-digit number)
35856742597589263259…66668402770959061130
Discovered Prime Numbers
p_k = 2^k Γ— origin + 1

These are the actual prime numbers discovered by this block, computed using the verified Primecoin formula. Each number has been independently confirmed to pass the Fermat primality test. Use the FactorDB links to verify any number independently.

1
origin + 1
3.585 Γ— 10⁹⁴(95-digit number)
35856742597589263259…66668402770959061131
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
2
2^1 Γ— origin + 1
7.171 Γ— 10⁹⁴(95-digit number)
71713485195178526518…33336805541918122261
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
3
2^2 Γ— origin + 1
1.434 Γ— 10⁹⁡(96-digit number)
14342697039035705303…66673611083836244521
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
4
2^3 Γ— origin + 1
2.868 Γ— 10⁹⁡(96-digit number)
28685394078071410607…33347222167672489041
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
5
2^4 Γ— origin + 1
5.737 Γ— 10⁹⁡(96-digit number)
57370788156142821214…66694444335344978081
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
6
2^5 Γ— origin + 1
1.147 Γ— 10⁹⁢(97-digit number)
11474157631228564242…33388888670689956161
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
7
2^6 Γ— origin + 1
2.294 Γ— 10⁹⁢(97-digit number)
22948315262457128485…66777777341379912321
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
8
2^7 Γ— origin + 1
4.589 Γ— 10⁹⁢(97-digit number)
45896630524914256971…33555554682759824641
Verify on FactorDB β†—Wolfram Alpha β†—

What this block proved

The miner who found this block proved the existence of 8 consecutive prime numbers forming a Cunningham Chain of the Second Kind. The prime chain origin β€” the large number shown above β€” anchors the chain and is divisible by a primorial (the product of small primes), cryptographically tying these prime numbers to this specific block.

View full Prime Chain Discovery page β†’
β˜…β˜†β˜†β˜†β˜†
Rarity
CommonChain length 8
How Primecoin's Proof-of-Work Constructs These Primes

Primecoin stores a value called the prime chain origin in each block. The miner found a large integer such that when divided by a primorial (the product of small primes: 2 Γ— 3 Γ— 5 Γ— 7 Γ— …), the result is the first prime in the chain. The origin is deliberately divisible by this primorial β€” that divisibility is part of the proof.

Prime Chain Origin = First Prime Γ— Primorial (2Β·3Β·5Β·7Β·11·…)
Source: Primecoin prime.cpp β€” CheckPrimeProofOfWork()

This is why the origin has many small prime factors β€” those factors are the primorial divisor. The chain then extends from the first prime using the 2CC formula:

2CC: p₁ (first prime), pβ‚‚ = 2p₁ βˆ’ 1, p₃ = 2pβ‚‚ βˆ’ 1, …
Verify on a Primecoin Node

Anyone running a Primecoin node can independently verify this block using the following RPC commands. Run these from the primecoin-cli command line or via the debug console in the Primecoin wallet.

1. Get block hash by height
getblockhash 31651

Returns the block hash for a given block height. Use this to confirm the hash shown above matches the chain.

2. Get full block data
getblock 3b15bf30ffb2ae5a905a7c4bd8a2b442e3fca4ea0c413633585c72e6efd2d1a3

Returns the full block header including difficulty, prime chain origin, prime chain type, transaction IDs, and all other fields shown on this page.

How to run these commands: To verify this data independently, open a terminal on your own Primecoin node and run primecoin-cli <command>. Alternatively, open the Primecoin-Qt wallet, go to Help β†’ Debug Window β†’ Console, and type the command directly. The node must be fully synced to this block height for the commands to return results.

Cross-reference on Chainz Explorer

Chainz is an independent Primecoin block explorer. Compare this block's data to verify accuracy.

View Block #31,651 on Chainz β†—
Circulating Supply:57,721,443 XPMΒ·at block #6,809,670 Β· updates every 60s
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AQZeJoQ81VgeHQBpEYTgvNqV1gnz3L5SvK

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