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Block #686,213

2CCLength 11β˜…β˜…β˜…β˜†β˜†

Cunningham Chain of the Second Kind Β· Discovered 8/21/2014, 11:22:46 AM Β· Difficulty 10.9540 Β· 6,122,176 confirmations

2CC
Cunningham Chain of the Second Kind

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

Block Header
Block Hash
aefe782abf11a7ea47eb92735c5e8fbf93078a3c9cdb9b9b50f0e3a50b3740f7
View on Chainz β†—

Height

#686,213

Difficulty

10.953956

Transactions

2

Size

14.01 KB

Version

2

Bits

0af43674

Nonce

38,517,500

Timestamp

8/21/2014, 11:22:46 AM

Confirmations

6,122,176

Mined by

⛏️ P2SHANSXnLY2HX2a5e6fdcDUZDtfwTSd25TjkD

Merkle Root

1d3fdcb3d46b114f87a2605795ce83c5f1d81d07c059f8b4ea7b2e7ae232f1fe
Transactions (2)
Coinbase14dc178d7668a0…c0850d12c42f2c
1 in β†’ 1 out8.5600 XPM116 B
ANSXnLY2…Sd25TjkD
988bf3e57c3c76…ab620640576ddc
95 in β†’ 2 out20000.0100 XPM13.80 KB
ARycv6E1…FrbVBCaDAe5WFHbn…gBo7rvHr
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.

2.263 Γ— 10⁹⁴(95-digit number)
22638709969167591865…83542336483393071181
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
2.263 Γ— 10⁹⁴(95-digit number)
22638709969167591865…83542336483393071181
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
2
2^1 Γ— origin + 1
4.527 Γ— 10⁹⁴(95-digit number)
45277419938335183730…67084672966786142361
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
3
2^2 Γ— origin + 1
9.055 Γ— 10⁹⁴(95-digit number)
90554839876670367460…34169345933572284721
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
4
2^3 Γ— origin + 1
1.811 Γ— 10⁹⁡(96-digit number)
18110967975334073492…68338691867144569441
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
5
2^4 Γ— origin + 1
3.622 Γ— 10⁹⁡(96-digit number)
36221935950668146984…36677383734289138881
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
6
2^5 Γ— origin + 1
7.244 Γ— 10⁹⁡(96-digit number)
72443871901336293968…73354767468578277761
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
7
2^6 Γ— origin + 1
1.448 Γ— 10⁹⁢(97-digit number)
14488774380267258793…46709534937156555521
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
8
2^7 Γ— origin + 1
2.897 Γ— 10⁹⁢(97-digit number)
28977548760534517587…93419069874313111041
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
9
2^8 Γ— origin + 1
5.795 Γ— 10⁹⁢(97-digit number)
57955097521069035174…86838139748626222081
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
10
2^9 Γ— origin + 1
1.159 Γ— 10⁹⁷(98-digit number)
11591019504213807034…73676279497252444161
Verify on FactorDB β†—Wolfram Alpha β†—
Γ—2βˆ’1 β†’
11
2^10 Γ— origin + 1
2.318 Γ— 10⁹⁷(98-digit number)
23182039008427614069…47352558994504888321
Verify on FactorDB β†—Wolfram Alpha β†—

What this block proved

The miner who found this block proved the existence of 11 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.

β˜…β˜…β˜…β˜†β˜†
Rarity
RareChain length 11

Approximately 1 in 1,000 blocks. Noteworthy discoveries.

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, …
Cross-reference on Chainz Explorer
Circulating Supply:57,711,167 XPMΒ·at block #6,808,388 Β· updates every 60s
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