Block #331,697

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/27/2013, 1:20:19 PM · Difficulty 10.1692 · 6,477,576 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

dc1aa4785c3e6fdb7feed130e1fec1d31fe717eed33b30770c297ab2a0f3f016

View on Chainz ↗

Height

#331,697

Difficulty

10.169229

Transactions

Size

1.79 KB

Version

Bits

0a2b5290

Nonce

6,760

Timestamp

12/27/2013, 1:20:19 PM

Confirmations

6,477,576

Mined by

⛏️ aR~AFutDVqJywBDvDDzwUNgUinUiETJTJj6UF

Merkle Root

5c3694c34aadbb8e96878cb66c0305f391c4f5ae5db8c647ca4e62dcfb697900

Transactions (4)

Coinbase14741535ca9435…539c84d1f4050d

1 in → 28 out9.6400 XPM1015 B

AFutDVqJ…TJTJj6UF AQ8QAT3J…rCFKuVbR Aac9Hjdg…P4ktypc3 AG5YAjec…VhA4Aeh1 AWCVEa7p…3vPBmiA2

57ae57ed10d8ed…8b84bfc63256bd

1 in → 1 out10.2100 XPM158 B

AZHoyXaw…uR4n6FYy

c0b0271389d526…d8472e1d3ae350

2 in → 2 out10.1647 XPM374 B

AMiLamm4…QZgbENNL AKH3Kyz4…cbY9pVtN

cda047b60824ac…499b7ccee5059c

1 in → 1 out0.2900 XPM191 B

AdkVwYi5…hr3oakWe

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.

1.021 × 10⁹⁵(96-digit number)

10211305720283096090…39966293869696540719

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

1.021 × 10⁹⁵(96-digit number)

10211305720283096090…39966293869696540719

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.021 × 10⁹⁵(96-digit number)

10211305720283096090…39966293869696540721

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

2.042 × 10⁹⁵(96-digit number)

20422611440566192181…79932587739393081439

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.042 × 10⁹⁵(96-digit number)

20422611440566192181…79932587739393081441

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

4.084 × 10⁹⁵(96-digit number)

40845222881132384363…59865175478786162879

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

4.084 × 10⁹⁵(96-digit number)

40845222881132384363…59865175478786162881

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

8.169 × 10⁹⁵(96-digit number)

81690445762264768726…19730350957572325759

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

8.169 × 10⁹⁵(96-digit number)

81690445762264768726…19730350957572325761

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

1.633 × 10⁹⁶(97-digit number)

16338089152452953745…39460701915144651519

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.633 × 10⁹⁶(97-digit number)

16338089152452953745…39460701915144651521

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 #331,697

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