Block #164,191

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 9/14/2013, 12:42:52 PM · Difficulty 9.8617 · 6,648,623 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

b25ea793c3799b782cc026753b45ce5f5416c1d580120b58c99652e2c6638ec4

View on Chainz ↗

Height

#164,191

Difficulty

9.861689

Transactions

Size

650 B

Version

Bits

09dc97aa

Nonce

157,318

Timestamp

9/14/2013, 12:42:52 PM

Confirmations

6,648,623

Mined by

⛏️ P2SHAFotq1BJb7R2Tqfs32RHm9xAg7P8SjWKWC

Merkle Root

a5c87034a47cf9a2f2a46dd1826fa9510498dbaf082e389931cf1b2fa6122df8

Transactions (3)

Coinbasef676a02a9546bf…fbca4ae1bbf5fb

1 in → 1 out10.2900 XPM109 B

AFotq1BJ…P8SjWKWC

1be2eaef4b30da…cfd8ff47e9bff4

1 in → 3 out10.2600 XPM225 B

ALMFFej6…53cKiC1R AYwTRdfJ…3whzbfUq AYnRf3yQ…bKcUQZf4

b589c3a71b3bce…9c6317ee49865d

1 in → 3 out10.2600 XPM226 B

AYVCr2pT…C3dQHPPa AHRnoPQw…SLFKBYxM AcyPVp7N…dDULWcdA

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.

7.725 × 10⁹³(94-digit number)

77250718649153108211…06859369153157457399

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

7.725 × 10⁹³(94-digit number)

77250718649153108211…06859369153157457399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

7.725 × 10⁹³(94-digit number)

77250718649153108211…06859369153157457401

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

1.545 × 10⁹⁴(95-digit number)

15450143729830621642…13718738306314914799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.545 × 10⁹⁴(95-digit number)

15450143729830621642…13718738306314914801

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

3.090 × 10⁹⁴(95-digit number)

30900287459661243284…27437476612629829599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.090 × 10⁹⁴(95-digit number)

30900287459661243284…27437476612629829601

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

6.180 × 10⁹⁴(95-digit number)

61800574919322486569…54874953225259659199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.180 × 10⁹⁴(95-digit number)

61800574919322486569…54874953225259659201

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

1.236 × 10⁹⁵(96-digit number)

12360114983864497313…09749906450519318399

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 #164,191

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