Block #491,142

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 4/14/2014, 8:00:50 AM · Difficulty 10.6800 · 6,326,814 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

45f6f14f33f05d04163b3d5080b40bf8ee65b272e27755b3282d2e4adb0dd846

View on Chainz ↗

Height

#491,142

Difficulty

10.679967

Transactions

Size

1.23 KB

Version

Bits

0aae124e

Nonce

47,907

Timestamp

4/14/2014, 8:00:50 AM

Confirmations

6,326,814

Mined by

⛏️ P2SHAeKNrjNjtSncLJjUrDnQbrfzMS1NEq95Ta

Merkle Root

2f5110df957049912f2da5d90c6980052e780ec08a9bfef0c01de9ac306b02a0

Transactions (5)

Coinbaseb6564413555a32…170de48c721cb5

1 in → 1 out8.7900 XPM110 B

AeKNrjNj…1NEq95Ta

2aa1c6c0f18817…1d3e90bf738b46

2 in → 2 out16.4594 XPM375 B

AGXp2oQN…nd21v9xA ASAk5QVj…qmoG5526

219f656d8f7dab…6a0c920f5bf54d

1 in → 2 out8.8000 XPM226 B

AGJkwG6P…AaFRcf3P AQMeVKux…Sq4pVZoq

77f536d481fa4c…48f3a63b0dfff3

1 in → 2 out8.8000 XPM227 B

AGFyKJMC…UHLroc9B AH4cnyxU…g4ZtWgik

dc547bafb5a057…d6388007846ed8

1 in → 2 out8.8000 XPM225 B

AMdkZdpH…aiMe3RNJ ALsuz559…Qqx4TbJe

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.

1.318 × 10¹⁰¹(102-digit number)

13187974296393225641…43500853167287036159

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

1.318 × 10¹⁰¹(102-digit number)

13187974296393225641…43500853167287036159

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.318 × 10¹⁰¹(102-digit number)

13187974296393225641…43500853167287036161

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

2.637 × 10¹⁰¹(102-digit number)

26375948592786451282…87001706334574072319

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.637 × 10¹⁰¹(102-digit number)

26375948592786451282…87001706334574072321

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

5.275 × 10¹⁰¹(102-digit number)

52751897185572902564…74003412669148144639

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

5.275 × 10¹⁰¹(102-digit number)

52751897185572902564…74003412669148144641

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

1.055 × 10¹⁰²(103-digit number)

10550379437114580512…48006825338296289279

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.055 × 10¹⁰²(103-digit number)

10550379437114580512…48006825338296289281

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

2.110 × 10¹⁰²(103-digit number)

21100758874229161025…96013650676592578559

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

2.110 × 10¹⁰²(103-digit number)

21100758874229161025…96013650676592578561

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 #491,142

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