Block #267,218

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 11/21/2013, 2:41:18 AM · Difficulty 9.9593 · 6,537,749 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

b0c785e8e75c864072dc58d481bebf3ee061875296ec7215b08c2210791b7833

View on Chainz ↗

Height

#267,218

Difficulty

9.959341

Transactions

Size

2.02 KB

Version

Bits

09f5975f

Nonce

11,292

Timestamp

11/21/2013, 2:41:18 AM

Confirmations

6,537,749

Mined by

⛏️ P2SHAeKNrjNjtSncLJjUrDnQbrfzMS1NEq95Ta

Merkle Root

8d93bea1f88da4288c66d1d039b0d8b9f946786c585c60ab64264a27d49e361b

Transactions (5)

Coinbase520399e0fe7422…e4b3adfef12430

1 in → 1 out10.1200 XPM110 B

AeKNrjNj…1NEq95Ta

bda7d0a79d5f68…3cfcf288311c9a

8 in → 2 out756.5373 XPM1.23 KB

AXgoyaZE…UMFu7gxX AXhwuRKV…9u2JSfYL

0c1f7828eabc24…b9a7554cfcaea5

1 in → 1 out10.0500 XPM158 B

ANPZmotn…Vo7H4mPW

96c57f9a5be064…e4d793d421625d

1 in → 3 out10.0500 XPM225 B

APSoHXZY…kNZx697r AcADmAoe…8iNnoMzB AeKNrjNj…1NEq95Ta

d4f3cfe6092ac2…0c4089929a14da

1 in → 2 out1.9925 XPM226 B

AZAGtx3d…zGQTU9Z1 ATKPWn12…NYghEd98

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.

5.744 × 10⁹⁵(96-digit number)

57445725533723204134…40424850590258329599

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

5.744 × 10⁹⁵(96-digit number)

57445725533723204134…40424850590258329599

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.744 × 10⁹⁵(96-digit number)

57445725533723204134…40424850590258329601

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

1.148 × 10⁹⁶(97-digit number)

11489145106744640826…80849701180516659199

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.148 × 10⁹⁶(97-digit number)

11489145106744640826…80849701180516659201

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

2.297 × 10⁹⁶(97-digit number)

22978290213489281653…61699402361033318399

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.297 × 10⁹⁶(97-digit number)

22978290213489281653…61699402361033318401

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

4.595 × 10⁹⁶(97-digit number)

45956580426978563307…23398804722066636799

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.595 × 10⁹⁶(97-digit number)

45956580426978563307…23398804722066636801

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

9.191 × 10⁹⁶(97-digit number)

91913160853957126615…46797609444133273599

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

9.191 × 10⁹⁶(97-digit number)

91913160853957126615…46797609444133273601

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