Block #214,154

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/17/2013, 7:42:55 AM · Difficulty 9.9228 · 6,601,971 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

ab45af0e66348148dbf5ed916dce2554bee755d561b4cab51498c375ff1ef5c4

View on Chainz ↗

Height

#214,154

Difficulty

9.922766

Transactions

Size

4.70 KB

Version

Bits

09ec3a63

Nonce

1,164,808,703

Timestamp

10/17/2013, 7:42:55 AM

Confirmations

6,601,971

Mined by

⛏️ DR_AZ4gvP32JykZfxFxsUJ5CdKrA4ukKsCiYr

Merkle Root

ca19e2b61a3766e1eb9ffafc92c3dd6d7a91898825a6e72d4ad4a8f5bb35686a

Transactions (1)

Coinbaseca19e2b61a3766…d4a8f5bb35686a

1 in → 137 out10.0900 XPM4.61 KB

AZ4gvP32…ukKsCiYr AReBFi8Q…Qq1cm1uS Aau9Lf88…RBtXCd78 AU3PaCwF…92DCmeEV AKwqFUdz…D4jGNKFN

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.

3.092 × 10⁹⁶(97-digit number)

30926769946115664052…54053172295905822399

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

3.092 × 10⁹⁶(97-digit number)

30926769946115664052…54053172295905822399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.092 × 10⁹⁶(97-digit number)

30926769946115664052…54053172295905822401

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

6.185 × 10⁹⁶(97-digit number)

61853539892231328104…08106344591811644799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

6.185 × 10⁹⁶(97-digit number)

61853539892231328104…08106344591811644801

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

1.237 × 10⁹⁷(98-digit number)

12370707978446265620…16212689183623289599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.237 × 10⁹⁷(98-digit number)

12370707978446265620…16212689183623289601

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

2.474 × 10⁹⁷(98-digit number)

24741415956892531241…32425378367246579199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.474 × 10⁹⁷(98-digit number)

24741415956892531241…32425378367246579201

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

4.948 × 10⁹⁷(98-digit number)

49482831913785062483…64850756734493158399

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 #214,154

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