Block #264,172

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/18/2013, 11:10:32 AM · Difficulty 9.9649 · 6,552,415 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

0fff9e28d10c610ead68c14015cea9fd32418f6b23debabb9f36164ceebd8edd

View on Chainz ↗

Height

#264,172

Difficulty

9.964912

Transactions

Size

228 B

Version

Bits

09f7047d

Nonce

14,026

Timestamp

11/18/2013, 11:10:32 AM

Confirmations

6,552,415

Mined by

⛏️ P2SHAKcbk49N7DP3nse7MJr3Ed6rZiGJbKa7j1

Merkle Root

350c0a9e1c5dfa3cee1322b2c9c002901ba03c2cb6ae14b19d5310b39d99c7f2

Transactions (1)

Coinbase350c0a9e1c5dfa…5310b39d99c7f2

1 in → 2 out10.0600 XPM137 B

AKcbk49N…GJbKa7j1 ALfEGCwh…VawujfMm

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.

2.714 × 10⁹⁶(97-digit number)

27149144940752775756…61654648836775747519

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

2.714 × 10⁹⁶(97-digit number)

27149144940752775756…61654648836775747519

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.714 × 10⁹⁶(97-digit number)

27149144940752775756…61654648836775747521

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

5.429 × 10⁹⁶(97-digit number)

54298289881505551513…23309297673551495039

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

5.429 × 10⁹⁶(97-digit number)

54298289881505551513…23309297673551495041

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

1.085 × 10⁹⁷(98-digit number)

10859657976301110302…46618595347102990079

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.085 × 10⁹⁷(98-digit number)

10859657976301110302…46618595347102990081

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

2.171 × 10⁹⁷(98-digit number)

21719315952602220605…93237190694205980159

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.171 × 10⁹⁷(98-digit number)

21719315952602220605…93237190694205980161

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

4.343 × 10⁹⁷(98-digit number)

43438631905204441211…86474381388411960319

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 #264,172

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