Block #163,266

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/13/2013, 9:26:22 PM · Difficulty 9.8614 · 6,645,832 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

ed0b98af9bd8a980aec5973b7743e23eb4900acb5bf7c6d44949b79ad3b87fd5

View on Chainz ↗

Height

#163,266

Difficulty

9.861427

Transactions

Size

572 B

Version

Bits

09dc8683

Nonce

246,081

Timestamp

9/13/2013, 9:26:22 PM

Confirmations

6,645,832

Mined by

⛏️ P2SHAaZgrM4iJMH6Vq7tSrHeh8fLLNAiWiChzm

Merkle Root

aafa697d599d87020f2a4a4ed0d7c7c93441c024a30b75eee266777fe2ebe6b4

Transactions (2)

Coinbase38a3efd9536686…5441fd45ec7381

1 in → 1 out10.2800 XPM109 B

AaZgrM4i…AiWiChzm

1db8e3f35b2d58…2d693183ddc025

2 in → 2 out250.0156 XPM374 B

APiS7kFM…1zEqfsSe AXCoN6AA…zcf98WDF

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.

7.966 × 10⁹¹(92-digit number)

79662351895068921676…69788260550799285939

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

7.966 × 10⁹¹(92-digit number)

79662351895068921676…69788260550799285939

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

7.966 × 10⁹¹(92-digit number)

79662351895068921676…69788260550799285941

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

1.593 × 10⁹²(93-digit number)

15932470379013784335…39576521101598571879

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.593 × 10⁹²(93-digit number)

15932470379013784335…39576521101598571881

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

3.186 × 10⁹²(93-digit number)

31864940758027568670…79153042203197143759

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.186 × 10⁹²(93-digit number)

31864940758027568670…79153042203197143761

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

6.372 × 10⁹²(93-digit number)

63729881516055137341…58306084406394287519

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.372 × 10⁹²(93-digit number)

63729881516055137341…58306084406394287521

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

1.274 × 10⁹³(94-digit number)

12745976303211027468…16612168812788575039

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 #163,266

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