Block #259,851

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/13/2013, 9:05:10 PM · Difficulty 9.9776 · 6,546,929 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

fd821e43ed4798306161826e359ec81f24c8427ddec668b9148a5f819df408bb

View on Chainz ↗

Height

#259,851

Difficulty

9.977590

Transactions

Size

946 B

Version

Bits

09fa4357

Nonce

379

Timestamp

11/13/2013, 9:05:10 PM

Confirmations

6,546,929

Mined by

⛏️ P2SHAKcbk49N7DP3nse7MJr3Ed6rZiGJbKa7j1

Merkle Root

515ba1c8cb1d8998eeb260d0d087d8217b39f559c03ddae9c3470d006c4bfb5e

Transactions (4)

Coinbase07f97e0215e374…b254890b7b3dd4

1 in → 2 out10.0600 XPM141 B

AKcbk49N…GJbKa7j1 ALfEGCwh…VawujfMm

9cb6c168f54833…549c45544b3f8c

1 in → 4 out10.0300 XPM261 B

ATJidoGN…etQegKbG AeKNrjNj…1NEq95Ta AQd7NAqj…KQ6cebzk AcADmAoe…8iNnoMzB

26d4a34c220e60…7b57573d225d6c

1 in → 2 out8.0094 XPM227 B

AYdVHCVT…uBUimwAG AUg6GXLW…xRwZLR65

e111b640ac4ebe…57cd8d7aae2bd2

1 in → 2 out2.8096 XPM227 B

AKh2ST8E…arq5QZFN AXQnio5G…cstPFJJr

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.

2.087 × 10⁹⁴(95-digit number)

20873600433677193029…36126783534565289719

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

2.087 × 10⁹⁴(95-digit number)

20873600433677193029…36126783534565289719

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.087 × 10⁹⁴(95-digit number)

20873600433677193029…36126783534565289721

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

4.174 × 10⁹⁴(95-digit number)

41747200867354386058…72253567069130579439

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.174 × 10⁹⁴(95-digit number)

41747200867354386058…72253567069130579441

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

8.349 × 10⁹⁴(95-digit number)

83494401734708772117…44507134138261158879

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.349 × 10⁹⁴(95-digit number)

83494401734708772117…44507134138261158881

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

1.669 × 10⁹⁵(96-digit number)

16698880346941754423…89014268276522317759

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.669 × 10⁹⁵(96-digit number)

16698880346941754423…89014268276522317761

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

3.339 × 10⁹⁵(96-digit number)

33397760693883508846…78028536553044635519

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 #259,851

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