Block #51,745

TWNLength 8★☆☆☆☆

Bi-Twin Chain · Discovered 7/16/2013, 7:09:46 AM · Difficulty 8.9029 · 6,764,461 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

af23c967d785de02d5d8d953c06c93389ebc05fb14a35ce9135b26c5dfe119c6

View on Chainz ↗

Height

#51,745

Difficulty

8.902875

Transactions

Size

204 B

Version

Bits

08e722d8

Nonce

224

Timestamp

7/16/2013, 7:09:46 AM

Confirmations

6,764,461

Mined by

⛏️ P2SHAJbRF9Vh3Pm1qDnFdVGkh4jvYocmDMZyVK

Merkle Root

82ec19c0a66c0bf35617e0c917390f4ab92fc791da1bae4e9030d731fad047d7

Transactions (1)

Coinbase82ec19c0a66c0b…30d731fad047d7

1 in → 1 out12.6000 XPM109 B

AJbRF9Vh…cmDMZyVK

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.

2.344 × 10¹⁰⁷(108-digit number)

23443689182239810673…31383915676433268759

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

2.344 × 10¹⁰⁷(108-digit number)

23443689182239810673…31383915676433268759

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.344 × 10¹⁰⁷(108-digit number)

23443689182239810673…31383915676433268761

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

4.688 × 10¹⁰⁷(108-digit number)

46887378364479621347…62767831352866537519

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.688 × 10¹⁰⁷(108-digit number)

46887378364479621347…62767831352866537521

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

9.377 × 10¹⁰⁷(108-digit number)

93774756728959242694…25535662705733075039

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

9.377 × 10¹⁰⁷(108-digit number)

93774756728959242694…25535662705733075041

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

1.875 × 10¹⁰⁸(109-digit number)

18754951345791848538…51071325411466150079

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.875 × 10¹⁰⁸(109-digit number)

18754951345791848538…51071325411466150081

Verify on FactorDB ↗Wolfram Alpha ↗

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

What this block proved

The miner who found this block proved the existence of 8 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 8

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 #51,745

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