Block #266,997

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/20/2013, 9:35:53 PM · Difficulty 9.9599 · 6,545,704 confirmations

TWN

Bi-Twin Chain

Interleaved pairs of primes that differ by 2, forming twin prime pairs at each level.

Block Header

Block Hash

ad6a8efe355e3936b5fad9e6083d3e120b3ee76a1d81ee79c0a4b489b9c6c909

View on Chainz ↗

Height

#266,997

Difficulty

9.959950

Transactions

Size

458 B

Version

Bits

09f5bf48

Nonce

81,963

Timestamp

11/20/2013, 9:35:53 PM

Confirmations

6,545,704

Mined by

⛏️ P2SHAeKNrjNjtSncLJjUrDnQbrfzMS1NEq95Ta

Merkle Root

3351dedd9c0961c0acb25cea7c45800ed86eb4b433579f0b78596b8f2561b1ac

Transactions (2)

Coinbase16d8c96587059c…bc406f1e28ac4b

1 in → 1 out10.0800 XPM110 B

AeKNrjNj…1NEq95Ta

7c349556107c0f…3fd3865704e67a

1 in → 4 out10.0400 XPM260 B

AeKNrjNj…1NEq95Ta APGLBd6z…qkbzdNLj AWP3nWpc…wKY1EyHx AcADmAoe…8iNnoMzB

Prime Chain Origin

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

1.562 × 10⁸⁹(90-digit number)

15629382138344893248…51234715793029665919

Discovered Prime Numbers

Lower: 2^k × origin − 1 | Upper: 2^k × 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.

Level 0 — Twin Prime Pair (origin ± 1)

origin − 1

1.562 × 10⁸⁹(90-digit number)

15629382138344893248…51234715793029665919

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.562 × 10⁸⁹(90-digit number)

15629382138344893248…51234715793029665921

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: origin + 1 − origin − 1 = 2 (twin primes ✓)

Level 1 — Twin Prime Pair (2^1 × origin ± 1)

2^1 × origin − 1

3.125 × 10⁸⁹(90-digit number)

31258764276689786496…02469431586059331839

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.125 × 10⁸⁹(90-digit number)

31258764276689786496…02469431586059331841

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^1 × origin + 1 − 2^1 × origin − 1 = 2 (twin primes ✓)

Level 2 — Twin Prime Pair (2^2 × origin ± 1)

2^2 × origin − 1

6.251 × 10⁸⁹(90-digit number)

62517528553379572992…04938863172118663679

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.251 × 10⁸⁹(90-digit number)

62517528553379572992…04938863172118663681

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^2 × origin + 1 − 2^2 × origin − 1 = 2 (twin primes ✓)

Level 3 — Twin Prime Pair (2^3 × origin ± 1)

2^3 × origin − 1

1.250 × 10⁹⁰(91-digit number)

12503505710675914598…09877726344237327359

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.250 × 10⁹⁰(91-digit number)

12503505710675914598…09877726344237327361

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^3 × origin + 1 − 2^3 × origin − 1 = 2 (twin primes ✓)

Level 4 — Twin Prime Pair (2^4 × origin ± 1)

2^4 × origin − 1

2.500 × 10⁹⁰(91-digit number)

25007011421351829196…19755452688474654719

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 9 consecutive prime numbers forming a Bi-Twin Chain. The prime chain origin — the large number shown above — anchors the chain and is divisible by a primorial (the product of small primes), cryptographically tying these prime numbers to this specific block.

★☆☆☆☆

Rarity

CommonChain length 9

Found in most blocks. The baseline for Primecoin's proof-of-work.

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 × 3 × 5 × 7 × …), the result is the first prime in the chain. The origin is deliberately divisible by this primorial — that divisibility is part of the proof.

Prime Chain Origin = First Prime × Primorial (2·3·5·7·11·…)

Source: Primecoin prime.cpp — CheckPrimeProofOfWork()

This is why the origin has many small prime factors — those factors are the primorial divisor. The chain then extends from the first prime using the TWN formula:

TWN: twin pairs (p, p+2) where p = origin/primorial − 1 and p+2 = origin/primorial + 1

Cross-reference on Chainz Explorer

Block #266,997

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