Block #269,756

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/23/2013, 10:30:22 AM · Difficulty 9.9523 · 6,542,886 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e8e8c7d3d6a028c6b3c4d0aca1273e88c50901990229029a5d6e150070fea440

View on Chainz ↗

Height

#269,756

Difficulty

9.952296

Transactions

Size

1.41 KB

Version

Bits

09f3c9a6

Nonce

120,752

Timestamp

11/23/2013, 10:30:22 AM

Confirmations

6,542,886

Mined by

⛏️ aR|AbRu7HeHQ3Yak9dDUFRUapQJUyhWbj5qBk

Merkle Root

ac14c864fc8053cd9392cf808778942ebf78c9dde86f840cb94f3799f522e5ab

Transactions (1)

Coinbaseac14c864fc8053…4f3799f522e5ab

1 in → 38 out10.0600 XPM1.32 KB

AbRu7HeH…hWbj5qBk APVGazMT…cDCk2nwA Aaf3CsqS…k1Eu3vmJ AMbCuLHZ…WSoStwkc AKfwt5JY…jsjT5QJQ

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.

2.475 × 10⁹⁹(100-digit number)

24752945707182404684…25824919945901291519

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

2.475 × 10⁹⁹(100-digit number)

24752945707182404684…25824919945901291519

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.475 × 10⁹⁹(100-digit number)

24752945707182404684…25824919945901291521

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

4.950 × 10⁹⁹(100-digit number)

49505891414364809369…51649839891802583039

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.950 × 10⁹⁹(100-digit number)

49505891414364809369…51649839891802583041

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

9.901 × 10⁹⁹(100-digit number)

99011782828729618739…03299679783605166079

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

9.901 × 10⁹⁹(100-digit number)

99011782828729618739…03299679783605166081

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

1.980 × 10¹⁰⁰(101-digit number)

19802356565745923747…06599359567210332159

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.980 × 10¹⁰⁰(101-digit number)

19802356565745923747…06599359567210332161

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

3.960 × 10¹⁰⁰(101-digit number)

39604713131491847495…13198719134420664319

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 #269,756

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