Block #312,391

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/15/2013, 12:05:44 AM · Difficulty 9.9958 · 6,491,137 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

b7b9f61ffeeecb0345e7485f1e86d80e709cbbf87757895ca533b4a984f26402

View on Chainz ↗

Height

#312,391

Difficulty

9.995801

Transactions

Size

869 B

Version

Bits

09feecd0

Nonce

80,518

Timestamp

12/15/2013, 12:05:44 AM

Confirmations

6,491,137

Mined by

⛏️ P2SHAeKNrjNjtSncLJjUrDnQbrfzMS1NEq95Ta

Merkle Root

eda4309c3289d68ea142122d00f07bbf8ba5d8f6f5f4bcd6384d63494cec3c0f

Transactions (2)

Coinbase6d9e884151c298…173c41a5987beb

1 in → 1 out10.0000 XPM110 B

AeKNrjNj…1NEq95Ta

66a21f39fe8e6c…05d83e1d9a281d

4 in → 2 out1.1475 XPM668 B

AYWRFUvM…x7bwVE4c AeaYHHSp…p95h5oia

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.

5.233 × 10⁹⁷(98-digit number)

52339806431644234772…98939793639374295039

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

5.233 × 10⁹⁷(98-digit number)

52339806431644234772…98939793639374295039

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.233 × 10⁹⁷(98-digit number)

52339806431644234772…98939793639374295041

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

1.046 × 10⁹⁸(99-digit number)

10467961286328846954…97879587278748590079

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.046 × 10⁹⁸(99-digit number)

10467961286328846954…97879587278748590081

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

2.093 × 10⁹⁸(99-digit number)

20935922572657693908…95759174557497180159

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.093 × 10⁹⁸(99-digit number)

20935922572657693908…95759174557497180161

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

4.187 × 10⁹⁸(99-digit number)

41871845145315387817…91518349114994360319

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.187 × 10⁹⁸(99-digit number)

41871845145315387817…91518349114994360321

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

8.374 × 10⁹⁸(99-digit number)

83743690290630775635…83036698229988720639

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

8.374 × 10⁹⁸(99-digit number)

83743690290630775635…83036698229988720641

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 10 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

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

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