Block #251,527

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/9/2013, 2:01:43 AM · Difficulty 9.9700 · 6,554,723 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

d6371dc55c454982d6f450e70420cbd0748c0ba40a81d1c26dd42e2a98cf66bd

View on Chainz ↗

Height

#251,527

Difficulty

9.969975

Transactions

Size

1.84 KB

Version

Bits

09f8504c

Nonce

18,139

Timestamp

11/9/2013, 2:01:43 AM

Confirmations

6,554,723

Mined by

⛏️ aR}xAPHngwz27GMW45jJC6wSAHExgSoXWG6CFr

Merkle Root

541ba2cb99a01beefa2978b311da600e2da926f74aacc77c5960d36da9149e18

Transactions (1)

Coinbase541ba2cb99a01b…60d36da9149e18

1 in → 51 out10.0300 XPM1.75 KB

APHngwz2…oXWG6CFr AL8BLb6A…ZXGDBoCK ALFWpHxd…zhX7LjhL AMttQDuf…7j6tfS9x ALzxhR4e…bD4bNcZe

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.

4.134 × 10⁹³(94-digit number)

41344937380477001271…31597299328089560959

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

4.134 × 10⁹³(94-digit number)

41344937380477001271…31597299328089560959

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.134 × 10⁹³(94-digit number)

41344937380477001271…31597299328089560961

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

8.268 × 10⁹³(94-digit number)

82689874760954002543…63194598656179121919

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.268 × 10⁹³(94-digit number)

82689874760954002543…63194598656179121921

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

1.653 × 10⁹⁴(95-digit number)

16537974952190800508…26389197312358243839

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.653 × 10⁹⁴(95-digit number)

16537974952190800508…26389197312358243841

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

3.307 × 10⁹⁴(95-digit number)

33075949904381601017…52778394624716487679

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.307 × 10⁹⁴(95-digit number)

33075949904381601017…52778394624716487681

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

6.615 × 10⁹⁴(95-digit number)

66151899808763202034…05556789249432975359

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 #251,527

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