Block #345,457

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 1/5/2014, 10:50:22 PM · Difficulty 10.2106 · 6,453,567 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

78a29456962375fadec0d6e812dddd1192e110371c249a5c372b877549b79ad4

View on Chainz ↗

Height

#345,457

Difficulty

10.210601

Transactions

Size

1.02 KB

Version

Bits

0a35e9f9

Nonce

16,369

Timestamp

1/5/2014, 10:50:22 PM

Confirmations

6,453,567

Mined by

⛏️ qEaReAQwRN8bEjE7sawFBq7N38V9niAV8PsTcY9

Merkle Root

93077db04091f7ed6200b3d999b0f9f068c4a4355dc810ed3d6ff5af486a20fe

Transactions (1)

Coinbase93077db04091f7…6ff5af486a20fe

1 in → 26 out9.5800 XPM947 B

AQwRN8bE…V8PsTcY9 AYtDvcGs…VuAcA7m7 AQ8QAT3J…rCFKuVbR Ad9pSzHt…cpJ3y2zz AN8nUzff…YcDfRDQ2

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.

4.105 × 10¹⁰¹(102-digit number)

41057075641240983137…95132180902019584639

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

4.105 × 10¹⁰¹(102-digit number)

41057075641240983137…95132180902019584639

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.105 × 10¹⁰¹(102-digit number)

41057075641240983137…95132180902019584641

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

8.211 × 10¹⁰¹(102-digit number)

82114151282481966275…90264361804039169279

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.211 × 10¹⁰¹(102-digit number)

82114151282481966275…90264361804039169281

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

1.642 × 10¹⁰²(103-digit number)

16422830256496393255…80528723608078338559

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.642 × 10¹⁰²(103-digit number)

16422830256496393255…80528723608078338561

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

3.284 × 10¹⁰²(103-digit number)

32845660512992786510…61057447216156677119

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.284 × 10¹⁰²(103-digit number)

32845660512992786510…61057447216156677121

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

6.569 × 10¹⁰²(103-digit number)

65691321025985573020…22114894432313354239

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

6.569 × 10¹⁰²(103-digit number)

65691321025985573020…22114894432313354241

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