Block #312,068

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/14/2013, 8:41:37 PM · Difficulty 9.9957 · 6,504,873 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

c9b37100e14fd3ee1a57b766e598565c51dbf4679ab79b889415250db2344983

View on Chainz ↗

Height

#312,068

Difficulty

9.995688

Transactions

Size

1.14 KB

Version

Bits

09fee570

Nonce

81,224

Timestamp

12/14/2013, 8:41:37 PM

Confirmations

6,504,873

Mined by

⛏️ aRoAac9Hjdgnw4T4L2csqLecJsmZjP4ktypc3

Merkle Root

b4b235e0ab3844f092adcf82196ada610c114167e75b0886f1540bfc157af9b7

Transactions (1)

Coinbaseb4b235e0ab3844…540bfc157af9b7

1 in → 30 out9.9700 XPM1.06 KB

Aac9Hjdg…P4ktypc3 AcM3ext6…Hwtj9Qp3 APeshfYV…3PKcKMKH AMSvYdDV…AM3cN4zw AbtgNzNb…9Atvu81w

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.

4.859 × 10⁹¹(92-digit number)

48591074687167715133…92072231876002389119

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

4.859 × 10⁹¹(92-digit number)

48591074687167715133…92072231876002389119

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.859 × 10⁹¹(92-digit number)

48591074687167715133…92072231876002389121

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

9.718 × 10⁹¹(92-digit number)

97182149374335430267…84144463752004778239

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

9.718 × 10⁹¹(92-digit number)

97182149374335430267…84144463752004778241

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

1.943 × 10⁹²(93-digit number)

19436429874867086053…68288927504009556479

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.943 × 10⁹²(93-digit number)

19436429874867086053…68288927504009556481

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

3.887 × 10⁹²(93-digit number)

38872859749734172107…36577855008019112959

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.887 × 10⁹²(93-digit number)

38872859749734172107…36577855008019112961

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

7.774 × 10⁹²(93-digit number)

77745719499468344214…73155710016038225919

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

7.774 × 10⁹²(93-digit number)

77745719499468344214…73155710016038225921

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 10 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

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

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 #312,068

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